text:
\section{Introduction}

The universal deformation and the simultaneous resolution of a simple
singularity are described by the corresponding simple Lie algebra
(Brieskorn \cite{bs:1}).
Inspired by that theory, the second named author
associated in \cite{sa:elliptic},\cite{sa:around}
a generalization of root systems,
consisting of vanishing cycles of the singularity,
to any regular weight systems \cite{sa:weight},
and asked to construct a suitable Lie theory
in order to reconstruct the primitive forms
for the singularities.
In fact, the simple singularities correspond exactly to
the weight systems having only positive exponents,
and, in this case,
this approach gives the classical finite root systems
as in \cite{bs:1}.
As the next case,
the approach is worked out for simple
elliptic singularities corresponding to weight
systems having only non-negative exponents,
from where the theory of elliptic Lie algebras
is emerging \cite{sa:elliptic}.
However, the root system in this approach
in general is hard to manipulate
because of the transcendental nature of vanishing cycles.
Hence,
he asked (\cite{sa:around}, Problem in p.124 in English version)
{\em an algebraic and/or a combinatorial
construction of the root system starting from a regular weight system}.

In \cite{t:2},
based on the mirror symmetry for the Landau-Ginzburg orbifolds
and also based on the duality theory of the weight systems
\cite{sa:duality},\cite{t:1},
the third named author proposed a new approach to the root systems,
answering to the above problem.
He introduced a triangulated category $D^b_{\mathbb Z}({\mathcal A}_f)$
of graded matrix factorizations
for a weighted homogeneous polynomial $f$
attached to a regular weight system
and showed that the category $D^b_{\mathbb Z}({\mathcal A}_f)$ for
a polynomial of type $A_l$
is equivalent to the bounded derived
category of modules over the path algebra of the Dynkin quiver
of type $A_l$.
He conjectured (\cite{t:2}, Conjecture 1.3)
further that the same type of equivalences hold
for all simple polynomials
of type ADE. The main goal of the present paper is to answer affirmatively
to the conjecture.

One side of this conjecture:
the properties of the category of
modules over a path algebra of a Dynkin quiver
are already well-understood
by the Gabriel's theorem \cite{g:1},
which states that the number of the indecomposable objects
in the category for a Dynkin quiver
coincides with the number of the positive roots of the root system
corresponding to the Dynkin diagram.
The other side of the conjecture:
the triangulated categories of (ungraded)
matrix factorizations were introduced and
developed by Eisenbud \cite{e:1} and Kn\"orrer \cite{k:1}
in the study of the maximal Cohen-Macaulay modules.
Recently, the categories of matrix factorizations are rediscovered
in string theory as the categories of topological D-branes of type B
in Landau-Ginzburg models (see \cite{kl:1},\cite{kl:2}).
The category $D^b_{\mathbb Z}({\mathcal A}_f)$ of graded matrix factorizations
is then motivated
by the work on the categories of topological D-branes of type B
in Landau-Ginzburg {\em orbifolds} $(f,{\mathbb Z}/h{\mathbb Z})$ by Hori-Walcher
\cite{hw:1}, where the orbifolding corresponds to introducing the
${\mathbb Q}$-grading.
In fact, in \cite{t:2},
the triangulated category $D^b_{\mathbb Z}({\mathcal A}_f)$ is constructed
from a special $A_\infty$-category with ${\mathbb Q}$-grading
via the twisted complexes
in the sense of Bondal-Kapranov \cite{bk:1}.
Independently, D.~Orlov defines a triangulated category,
called the category of graded D-branes of type B,
which is in fact equivalent to $D^b_{\mathbb Z}({\mathcal A}_f)$
(see the end of subsection \ref{ssec:Dz}).
Though some notions, for instance the central charge of the stability
condition (see section \ref{sec:stab}),
can be understood more naturally in $D^b_{\mathbb Z}({\mathcal A}_f)$,
the Orlov's construction of categories
requires less terminologies and is easier to understand
in a traditional way in algebraic geometry.
Therefore, in this paper
we shall use the Orlov's construction with a slight modification
of the scaling of degrees and denote
the modified category by $\tHMF{R}$.

Let us explain details of the contents of the present paper.
In section \ref{sec:cat}, we recall the construction of
triangulated categories of matrix factorizations.
Since we compare the category $HMF_R^{gr}(f)$
with the ungraded version $HMF_{\mathcal O}(f)$
in the proof of our main theorem (Theorem \ref{thm:main}),
we first introduce the ungraded version
$HMF_{\mathcal O}(f)$ corresponding to that given in \cite{o:1}
in subsection \ref{ssec:D}, and then
we define the graded version $HMF_R^{gr}(f)$
based on \cite{o:2} in subsection \ref{ssec:Dz}, where
we also explain the relation of the category $HMF_R^{gr}(f)$ with
the category $D^b_{\mathbb Z}({\mathcal A}_f)$ introduced in \cite{t:2}.
Section \ref{sec:main} is the main part of the present paper.
In subsection \ref{ssec:main-state}, we state the main theorem
(Theorem \ref{thm:main}):
for a polynomial $f$ of type ADE,
$HMF_R^{gr}(f)$ is equivalent as a triangulated category
to the bounded derived category of modules over the path algebra
of the Dynkin quiver of type of $f$.
Subsection \ref{ssec:proof} is devoted to the proof of
Theorem \ref{thm:main}.
The proof is based on various explicit data
on the matrix factorizations;
the complete list of the matrix factorizations
(Table \ref{tbl:1}), their gradings (Table \ref{tbl:2}) and
the complete list of the morphisms in $HMF_R^{gr}(f)$
(Table \ref{tbl:3}).
The tables are arranged in the final section (section \ref{sec:tbl}).
In section \ref{sec:stab},
we construct a stability condition,
the notion of which is introduced by Bridgeland \cite{bd:1},
for the triangulated category $HMF_R^{gr}(f)$.
One can see that, as in the $A_l$ case \cite{t:2},
the phase of objects
(see Theorem \ref{thm:grading} or Table \ref{tbl:2})
and the central charge ${\mathcal Z}$ (Definition \ref{defn:centralcharge})
can be naturally given by the grading of matrix factorizations
in Table \ref{tbl:2} (c.f. \cite{w:1}).
They in fact define a stability condition
on $HMF_R^{gr}(f)$ (Theorem \ref{thm:stab}),
from which an abelian category is obtained as
a full subcategory of $HMF_R^{gr}(f)$.
In Proposition \ref{prop:abelian}, we show
that this abelian category is equivalent to an abelian category of
modules over the path algebra ${\mathbb C}\vec{\Delta}_{principal}$,
where $\vec{\Delta}_{principal}$ is
the Dynkin quiver with the orientation being taken to be
the principal orientation introduced in \cite{sa:principal}.
In the appendix, we include another proof of Theorem \ref{thm:main}
by K.~Ueda.

{\bf Acknowledgement} :\
We are grateful to M.~Kashiwara and K.~Watanabe for valuable discussions.
This work was partly supported by Grant-in Aid for Scientific Research
grant numbers 16340016, 17654015 and 17740036 of the Ministry of Education,
Science and Culture in Japan.
H.~K is supported by JSPS Research Fellowships for Young
Scientists.

\section{Triangulated categories of matrix factorizations}
\label{sec:cat}

In this section, we set up several definitions which are used
in the present paper.
The goal of this section is the introduction of the categories
$\HMF{A}$ and $\tHMF{A}$ attached
to a weighted homogeneous polynomial $f\in A$
following \cite{o:1},\cite{o:2} with slight modifications.

 \subsection{The triangulated category ${\HMF{A}}$
of matrix factorizations} \hfill \\
\label{ssec:D}
Let $A$ be either the polynomial ring $R:={\mathbb C}[x,y,z]$,
the convergent power series ring ${\mathcal O}:={\mathbb C}\{x,y,z\}$ or
the formal power series ring ${\hat {\mathcal O}}:={\mathbb C}[[x,y,z]]$
in three variables $x,y$ and $z$.
\begin{defn}[Matrix factorization]
For a nonzero polynomial $f\in A$,
a {\em matrix factorization} $M$ of $f$ is defined by
\begin{equation*}
 M:=\Big( \mf{P_0}{p_0}{p_1}{P_1}\Big) \ ,
\end{equation*}
where $P_0$, $P_1$ are right {\em free}
$A$-modules of finite rank,
and $p_0:P_0\to P_1$, $p_1:P_1\to P_0$ are
$A$-homomorphisms such that $p_1p_0=f\cdot\mathrm{id}_{P_0}$ and
$p_0p_1=f\cdot\mathrm{id}_{P_1}$.
The set of all matrix factorizations of $f$
is denoted by $\MF{A}$.
 \label{defn:MF}
\end{defn}
Since $p_0p_1$ and $p_1p_0$ are
$f$ times
the identities,
where $f$ is nonzero element of $A$,
the rank of $P_0$ coincides with that of $P_1$.
We call the rank the {\em size} of the matrix factorization $M$.
\begin{defn}[Homomorphism]
Given two matrix factorizations
$M:=(\mfs{P_0}{p_0}{p_1}{P_1})$ and
$M':=(\mfs{P'_0}{p'_0}{p'_1}{P'_1})$,
a {\em homomorphism} $\Phi:M\to M'$
is a pair of $A$-homomorphisms $\Phi=(\phi_0,\phi_1)$
\begin{equation*}
 \phi_0:P_0\to P_0'\ ,\qquad \phi_1:P_1\to P_1'\ ,
\end{equation*}
such that the following diagram commutes:
\begin{equation*}
 \xymatrix{
 \ P_0\ \ar[r]^{p_0} \ar[d]^{\phi_0} &
 \ P_1\ \ar[r]^{p_1} \ar[d]^{\phi_1} &
 \ P_0\ \ar[d]^{\phi_0} & \\
 \ P'_0\ \ar[r]^{p'_0} &
 \ P'_1\ \ar[r]^{p'_1} &
 \ P'_0\ & \hspace*{-1.7cm} .
}
\end{equation*}
The set of all homomorphisms from $M$ to $M'$,
denoted by $\mathrm{Hom}_\MF{A}(M,M')$, is naturally an $A$-module and
is finitely generated,
since the sizes of the matrix factorizations are finite.
For three matrix factorizations $M$, $M'$, $M''$
and homomorphisms $\Phi:M\to M'$ and $\Phi':M'\to M''$,
the composition $\Phi'\Phi$ is defined by
\begin{equation*}
 \Phi'\Phi=(\phi'_0\phi_0,\phi_1'\phi_1)\ .
\end{equation*}
This composition
is associative: $\Phi''(\Phi'\Phi)=(\Phi''\Phi')\Phi$
for any three homomorphisms.
 \label{defn:hom}
\end{defn}
\begin{defn}[$\HMF{A}$]
An additive category $\HMF{A}$ is defined by the following data.
The set of objects is given by the set of all matrix factorizations:
\begin{equation*}
 \mathrm{Ob}(\HMF{A}):=\MF{A}\ .
\end{equation*}
For any two objects $M,M'\in\MF{A}$,
the set of morphisms is given by the quotient module:
\begin{equation*}
 \HomD{A}(M,M'):=
 \mathrm{Hom}_\MF{A}(M,M')/ \sim \ ,
\end{equation*}
where two elements $\Phi$, $\Phi'$ in $\mathrm{Hom}_\MF{A}(M,M')$ are equivalent
(homotopic)
$\Phi\sim\Phi'$ if there exists
a {\em homotopy} $(h_0,h_1)$, {\it i.e.},
a pair $(h_0,h_1):(P_0\to P_1',P_1\to P_0')$ of $A$-homomorphisms
such that $\Phi'-\Phi=(p_1'h_0+h_1p_0, p_0'h_1+h_0p_1)$.
The composition of morphisms on $\HomD{A}$ is induced from that
on $\mathrm{Hom}_\MF{A}$
since $\Phi\sim\Phi'$ and $\Psi\sim\Psi'$ imply $\Psi\Phi\sim\Psi'\Phi'$.
\end{defn}
Note that the matrix factorization $M=(\mfs{P_0}{p_0}{p_1}{P_1})\in\MF{A}$
of size one with $(p_0,p_1)=(1,f)$ or $(p_0,p_1)=(f,1)$
defines the zero object in $\HMF{A}$,
that is: one has
$\HomD{A}(M,M')=\HomD{A}(M',M)=0$
for any matrix factorization $M'\in\MF{A}$
or, equivalently, $\mathrm{id}_M\in\mathrm{Hom}_{\MF{A}}(M,M)$ is homotopic to zero.
\begin{lem}
For any two matrix factorizations $M,M'\in\HMF{A}$,
the space of morphisms $\HomD{A}(M,M')$ is a finitely generated
$A/\left(\frac{{\partial } f}{{\partial } x},\frac{{\partial } f}{{\partial } y},\frac{{\partial } f}{{\partial } z}\right)$-module.
 \label{lem:finite}
\end{lem}
\begin{pf}
Since
$\mathrm{Hom}_\MF{A}(M,M')$ is a finitely generated $A$-module and
the equivalence relation $\sim$ is given by
quotienting out by an $A$-submodule,
$\HomD{A}(M,M')$ is also a finitely generated $A$-module.
On the other hand, the Jacobi ideal
$\left(\frac{{\partial } f}{{\partial } x},\frac{{\partial } f}{{\partial } y},\frac{{\partial } f}{{\partial } z}\right)$
annihilates $\HomD{A}(M,M')$: that is,
$\frac{{\partial } f}{{\partial } x}\left(\phi_0,\phi_1\right)\sim
\frac{{\partial } f}{{\partial } y}\left(\phi_0,\phi_1\right)\sim
\frac{{\partial } f}{{\partial } z}\left(\phi_0,\phi_1\right)\sim 0$
for any morphism $\Phi=(\phi_0,\phi_1)$.
This can be shown for instance by
differentiating $p_1p_0=f\cdot\mathrm{id}_{P_0}$ and
$p_0p_1=f\cdot\mathrm{id}_{P_1}$
by $\fpartial{x}$. Then
we have two identities
$\frac{{\partial } p_1}{{\partial } x}p_0+p_1\frac{{\partial } p_0}{{\partial } x}=
\frac{{\partial } f}{{\partial } x}\cdot\mathrm{id}_{P_0}$
and
$\frac{{\partial } p_0}{{\partial } x}p_1+p_0\frac{{\partial } p_1}{{\partial } x}
=\frac{{\partial } f}{{\partial } x}\cdot\mathrm{id}_{P_1}$.
Multiplying $\phi_0$ and $\phi_1$ by these two identities, respectively,
leads to $\frac{{\partial } f}{{\partial } x}\left(\phi_0,\phi_1\right)\sim 0$,
where $(\phi_1\frac{{\partial } p_0}{{\partial } x},\phi_0\frac{{\partial } p_1}{{\partial } x})$
is the corresponding homotopy.
In a similar way one can obtain
$\frac{{\partial } f}{{\partial } y}\left(\phi_0,\phi_1\right)\sim
\frac{{\partial } f}{{\partial } z}\left(\phi_0,\phi_1\right)\sim 0$.
\qed\end{pf}
\begin{defn}[Shift functor]
The {\em shift functor} $T:\HMF{A}\to\HMF{A}$
is defined as follows.
The action of $T$ on
$M=(\mfs{P_0}{p_0}{p_1}{P_1})\in\HMF{A}$ is given by
\begin{equation*}
 T\Big( \mf{P_0}{p_0}{p_1}{P_1}\Big)
 :=\Big(\mf{P_1}{-p_1}{-p_0}{P_0}\Big)\ .
\end{equation*}
For any $M,M'\in\HMF{A}$, the action of $T$ on
$\Phi=(\phi_0,\phi_1)\in\HomD{A}(M,M')$ is given by
\begin{equation*}
 T(\phi_0,\phi_1):=(\phi_1,\phi_0)\ .
\end{equation*}
 \label{defn:shift}
\end{defn}
Note that
the square $T^2$ of the shift functor is isomorphic to the identity functor
on $\HMF{A}$.
\begin{defn}[Mapping cone]
For an element $\Phi=(\phi_0,\phi_1)\in\mathrm{Hom}_\MF{A}(M,M')$,
the {\em mapping cone} $C(\Phi)\in\MF{A}$ is defined by
\begin{equation*}
 \begin{split}
 & C(\Phi):=\Big( \mf{C_0}{c_0}{c_1}{C_1}\Big)\ ,\quad\text{where} \\
 & C_0:=P_1\oplus P_0'\ ,\qquad C_1:=P_0\oplus P_1'\ ,\qquad
   c_0:=\begin{pmatrix} -p_1 & 0 \\ \phi_1 & p_0'\end{pmatrix}\ ,\qquad
   c_1:=\begin{pmatrix} -p_0 & 0 \\ \phi_0 & p_1'\end{pmatrix}\ .
 \end{split}
\end{equation*}
 \label{defn:cone}
\end{defn}
The following is stated in \cite{o:1} Proposition 3.3.
\begin{prop}
The additive category $\HMF{A}$ endowed with the shift functor
$T$ and the distinguished triangles
forms a triangulated category,
where a distinguished triangle is
a sequence of morphisms which is isomorphic to the
sequence
\begin{equation*}
 M\stackrel{\Phi}{\to} M' \to C(\Phi)\to T(M)\
\end{equation*}
for some $M,M'\in\MF{A}$ and $\Phi\in\mathrm{Hom}_\MF{A}(M,M')$.
\end{prop}
\begin{pf}
The proof is
the same as the proof of the analogous
result for a usual homotopic category
(see e.g. \cite{gm:1}, \cite{ks:1}).
\qed\end{pf}

 \subsection{The triangulated category $\tHMF{R}$
of graded matrix factorizations}
\hfill\\
\label{ssec:Dz}
In this subsection,
we study graded matrix factorizations for
a weighted homogeneous polynomial $f$
and construct the corresponding triangulated category, denoted by
$\tHMF{R}$.

A quadruple $W:=(a,b,c;h)$ of positive integers
with $\mathrm{g.c.d}(a,b,c)=1$ is called
a {\em weight system}.
For a weight system $W$, we define the
{\em Euler vector field} $E=E_W$ by
\begin{equation*}
  E:= \frac{a}{h}x\frac{{\partial }}{{\partial } x}+\frac{b}{h}y\frac{{\partial }}{{\partial } y}+
 \frac{c}{h}z\frac{{\partial }}{{\partial } z}\ .
\end{equation*}
For a given weight system $W$,
${R}$ becomes a graded ring by putting
$\deg(x)=\frac{2a}{h}$, $\deg(y)=\frac{2b}{h}$ and
$\deg(z)=\frac{2c}{h}$.
Let ${R}=\oplus_{d\in\frac{2}{h}{\mathbb Z}_{\ge 0}} {R}_d$
be the graded piece decomposition,
where
${R}_d:=\{f\in R\ |\ 2Ef=df\}$\ .
A weight system $W$ is called {\em regular} (\cite{sa:weight})
if the following equivalent conditions are satisfied:
\begin{itemize}
 \item[(a)] $\chi_W(T):=
T^{-h}\frac{(T^h-T^a)(T^h-T^b)(T^h-T^c)}{(T^a-1)(T^b-1)(T^c-1)}$
has no poles except at $T=0$.

 \item[(b)]
A generic element of the eigenspace $R_2=\{f\in R\ |\ Ef= f\}$
has an isolated critical point at the origin,
{\it i.e.},
the Jacobi ring
$R/\left(\frac{{\partial } f}{{\partial } x},\frac{{\partial } f}{{\partial } y},\frac{{\partial } f}{{\partial } z}\right)$
is finite dimensional over ${\mathbb C}$.
\end{itemize}
Such an element $f$ of $R_2$ as in (b)
shall be called a {\em polynomial of type $W$}.

In the present paper,
{\em by a graded module, we mean a graded right module with degrees
only in $\frac{2}{h}{\mathbb Z}$}.
Namely, a graded ${R}$-module $\ti{P}$ decomposes
into the direct sum:
\begin{equation}
\ti{P}=\oplus_{d\in\frac{2}{h}{\mathbb Z}}\ti{P}_d\ .
 \label{P-grad}
\end{equation}
For two graded ${R}$-modules $\ti{P}$ and $\ti{P}'$,
a graded ${R}$-homomorphism $\phi$ of degree $s\in\frac{2}{h}{\mathbb Z}$
is an ${R}$-homomorphism $\phi:\ti{P}\to\ti{P}'$ such that
$\phi(\ti{P}_d)\subset\ti{P}'_{d+s}$ for any $d$.
The category of graded ${R}$-modules has a degree shifting
automorphism $\tau$ defined by
\footnote{This $\tau$ is what is often denoted
(for instance \cite{y},\cite{o:2}) by $(1)$,
{\it i.e.} , $\tau(\ti{P})=\ti{P}(1)$.
}
\begin{equation*}
 (\tau(\ti{P}))_d:=\ti{P}_{d+\frac{2}{h}}\ .
\end{equation*}
For any two graded ${R}$-modules $\ti{P}$, $\ti{P}'$
and a graded ${R}$-homomorphism $\phi:\ti{P}\to\ti{P}'$,
we denote the induced graded ${R}$-homomorphism by
$\tau(\phi):\tau(\ti{P})\to\tau(\ti{P}')$.
On the other hand, an ${R}$-homomorphism $\phi:\ti{P}\to\ti{P}'$
of degree $\frac{2m}{h}$ induces
a degree zero ${R}$-homomorphism from $\ti{P}$ to $\tau^m(\ti{P}')$,
which we denote again by $\phi:\ti{P}\to\tau^m(\ti{P}')$.
\begin{defn}[Graded matrix factorization]
For a
polynomial $f$ of type $W$,
a {\em graded matrix factorization} $\ti{M}$ of $f\in {R}$
is defined by
\begin{equation*}
 \ti{M}:=\Big( \mf{\ti{P}_0}{p_0}{p_1}{\ti{P}_1}\Big) \ ,
\end{equation*}
where $\ti{P}_0$, $\ti{P}_1$ are
free graded right ${R}$-modules of finite rank,
$p_0:\ti{P}_0\to\ti{P}_1$ is a graded ${R}$-homomorphism of degree zero,
$p_1:\ti{P}_1\to\ti{P}_0$ is a graded ${R}$-homomorphism of degree two
such that $p_1p_0=f\cdot\mathrm{id}_{\ti{P}_0}$ and $p_0p_1=f\cdot\mathrm{id}_{\ti{P}_1}$.
The set of all graded matrix factorizations of $f$
is denoted by $\tMF{R}$.
 \label{defn:gMF}
\end{defn}
\begin{defn}[Homomorphism]
Given two graded matrix factorizations
$\ti{M},\ti{M}'\in\tMF{R}$,
a {\em homomorphism}
$\Phi=(\phi_0,\phi_1):\ti{M}\to\ti{M}'$
is a homomorphism
in the sense of Definition \ref{defn:hom}
such that $\phi_0$ and $\phi_1$ are graded ${R}$-homomorphisms
of {\em degree zero}.
The vector space of all graded
${R}$-homomorphisms from $\ti{M}$ to $\ti{M}'$
is denoted by $\mathrm{Hom}_\tMF{R}(\ti{M},\ti{M}')$.
 \label{defn:gHom}
\end{defn}
For three graded matrix factorizations
$\ti{M},\ti{M}',\ti{M}''\in\tMF{R}$ and
morphisms $\Phi:\ti{M}\to\ti{M}'$,
$\Phi':\ti{M}'\to\ti{M}''$, the composition is again
a graded ${R}$-homomorphism:
$\Phi'\Phi\in\mathrm{Hom}_\tMF{R}(\ti{M},\ti{M}'')$.
\begin{defn}[$\tHMF{R}$]\label{defn:zderived}
An additive category
$\tHMF{R}$ of graded matrix factorizations is defined
by the following data.
The set of objects is given by the set of
all graded matrix factorizations:
\begin{equation*}
\mathrm{Ob}(\tHMF{R}):= \tMF{R}\ .
\end{equation*}
For any two objects $\ti{M},\ti{M}'\in\tMF{R}$,
the set of morphisms is given by
\begin{equation*}
\HomDz{R}(\ti{M},\ti{M}'):=\mathrm{Hom}_\tMF{R}(\ti{M},\ti{M}')/\sim\ ,
\end{equation*}
where two elements $\Phi$, $\Phi'$ in $\mathrm{Hom}_\tMF{R}(\ti{M},\ti{M}')$
are equivalent $\Phi\sim\Phi'$
if there exists a {\em homotopy}, {\it i.e.},
a pair $(h_0,h_1):(\ti{P}_0\to\ti{P}_1',\ti{P}_1\to\ti{P}_0')$
of graded $R$-homomorphisms
such that $h_0$ is of degree minus two, $h_1$ is of degree zero and
$\Phi'-\Phi=(\tau^{-h}(p_1')h_0+h_1p_0, p_0'h_1+\tau^h(h_0)p_1)$.
The composition of morphisms is induced from that
on $\mathrm{Hom}_\tMF{R}(\ti{M},\ti{M}')$.
 \label{defn:gD}
\end{defn}
A graded matrix factorization is
the zero object in $\tHMF{R}$
if and only if it is a direct sum of
the graded matrix factorizations of the forms
$(\mfs{\tau^n(R)}{1}{f}{\tau^n(R)})\in\tMF{R}$ and
$(\mfs{\tau^{n'}(R)}{f\ }{1\ }{\tau^{n'+h}(R)})\in\tMF{R}$
for some $n,n'\in{\mathbb Z}$.
\begin{lem}
The category $\tHMF{R}$ is Krull-Schmidt, that is, \\
(a)\ for any two objects $\ti{M},\ti{M}'\in\tHMF{R}$,
$\HomDz{R}(\ti{M},\ti{M}')$ is finite dimensional; \\
(b)\ for any object $\ti{M}\in\tHMF{R}$ and
any idempotent $e\in\HomDz{R}(\ti{M},\ti{M})$,
there exists a matrix factorization $\ti{M}'\in\tHMF{R}$
and a pair of morphisms
$\Phi\in\HomDz{R}(\ti{M},\ti{M}')$, $\Phi'\in\HomDz{R}(\ti{M}',\ti{M})$
such that $e=\Phi'\Phi$ and $\Phi\Phi'=\mathrm{id}_{\ti{M}'}$.
 \label{lem:KS}
\end{lem}
\begin{pf}
(a)\ Due to Lemma \ref{lem:finite},
$\oplus_{n\in{\mathbb Z}}\HomDz{R}(\ti{M},\tau^n(\ti{M}'))$
is a finitely generated graded
${R}/\left(\frac{{\partial } f}{{\partial } x},\frac{{\partial } f}{{\partial } y},
\frac{{\partial } f}{{\partial } z}\right)$-module.
Since the Jacobi ring
${R}/\left(\frac{{\partial } f}{{\partial } x},\frac{{\partial } f}{{\partial } y},\frac{{\partial } f}{{\partial } z}\right)$
is finite dimensional,
the space $\HomDz{R}(\ti{M},\ti{M}')$
is finite dimensional over ${\mathbb C}$.

(b)\ Let $R_+$ be the maximal ideal of $R$
of all positive degree elements.
Note that any graded matrix factorization is isomorphic
in $\tHMF{R}$ to a graded matrix factorization
whose entries belong to $\tau^n(R_+)$ for some $n\in{\mathbb Z}$.
Thus, we may assume that
$\ti{M}:=(\mfs{\ti{P}_0}{p_0}{p_1}{\ti{P}_1})\in\tHMF{R}$
is such a graded matrix factorization.
Suppose that $\ti{M}$ has an idempotent
$e\in\HomDz{R}(\ti{M},\ti{M})$, $e^2=e$. This implies that there exists
$\hat{e}\in\mathrm{Hom}_{\tMF{R}}(\ti{M},\ti{M})$ such that
\begin{equation}\label{KS}
 \hat{e}^2-\hat{e}=(\tau^{-h}(p_1)h_0+h_1p_0,p_0h_1+\tau^h(h_0)p_1)
\end{equation}
for some homotopy $(h_0,h_1)$ on $\HomDz{R}(\ti{M},\ti{M})$.
However, since each entry of $p_0$ and $p_1$ belongs to
$\tau^n(R_+)$, each entry in the right hand side also belongs to
$\tau^n(R_+)$. Let
$\pi:\mathrm{Hom}_{\tMF{R}}(\ti{M},\ti{M})\to\mathrm{Hom}_{\tMF{R}}(\ti{M},\ti{M})$
be the canonical projection given by restricting each entry on
$R/R_+={\mathbb C}$. Then, eq.(\ref{KS}) in fact implies that
$\pi(\hat{e})^2-\pi(\hat{e})=0$.
Thus, for $\pi(\hat{e})=:(\hat{e}_0,\hat{e}_1)$,
defining a matrix factorization $\ti{M}'\in\tHMF{R}$ by
\begin{equation*}
 \ti{M}':=\Big(
 \mfl{\hat{e}_0\ti{P}_0}{\hat{e}_1p_0\hat{e}_0}
{\hat{e}_0p_1\hat{e}_1}{\hat{e}_1\ti{P}_1}
 \Big) \ ,
\end{equation*}
one obtains a pair of morphisms
$\Phi\in\HomDz{R}(\ti{M},\ti{M}')$ and $\Phi'\in\HomDz{R}(\ti{M}',\ti{M})$
such that $e=\Phi'\Phi$ and $\Phi\Phi'=\mathrm{id}_{\ti{M}'}$.
\qed\end{pf}

One can see that $\tau$ induces an automorphism on $\tHMF{R}$,
which we shall denote by the same notation
$\tau:\tHMF{R}\to\tHMF{R}$.
Explicitly, the action of $\tau$ on
$\ti{M}=(\mfs{\ti{P}_0}{p_0}{p_1}{\ti{P}_1})\in\tHMF{R}$ is given by
\begin{equation*}
 \tau \Big(\mf{\ti{P}_0}{p_0}{p_1}{\ti{P}_1}\Big):=
 \Big(\mf{\tau(\ti{P}_0)}{\tau(p_0)}{\tau(p_1)}{\tau(\ti{P}_1)}\Big)\ .
\end{equation*}
The action of $\tau$ on morphisms are naturally induced from that
on graded ${R}$-homomorphisms between two graded right ${R}$-modules.

Also, we have the shift functor $T$ on $\HMF{R}$,
the graded version of that in Definition \ref{defn:shift}.
\footnote{The shift functor $T$ is often denoted by $[1]$. }
\begin{defn}[Shift functor on $\tHMF{R}$]
The {\it shift functor}
$T:\tHMF{R}\to\tHMF{R}$ is defined as follows.
The action of $T$ on $\ti{M}\in\tHMF{R}$ is given by
\begin{equation*}
 T\Big(\mf{\ti{P}_0}{p_0}{p_1}{\ti{P}_1}\Big)
 :=\Big(\mf{\ti{P}_1}{-p_1\quad}{-\tau^h(p_0)\quad}{\tau^h(\ti{P}_0)}\Big)\ .
\end{equation*}
For any $\ti{M},\ti{M}'\in\tHMF{R}$, the action of $T$ on
$\Phi=(\phi_0,\phi_1)\in\HomDz{R}(\ti{M},\ti{M}')$
is given by
\begin{equation*}
 T(\phi_0,\phi_1):=(\phi_1,\tau^h(\phi_0))\ .
\end{equation*}
 \label{defn:shift_Z}
\end{defn}
We remark that the square $T^2$ of the shift functor is not isomorphic
to the identity functor on $\tHMF{R}$ but $T^2=\tau^h$.
\begin{defn}[Mapping cone]
For an element $\Phi=(\phi_0,\phi_1)\in\mathrm{Hom}_\tMF{R}(\ti{M},\ti{M}')$,
the {\em mapping cone} $C(\Phi)\in\tMF{R}$ is defined by
\begin{equation*}
 \begin{split}
 & C(\Phi):=\Big( \mf{C_0}{c_0}{c_1}{C_1}\Big)\ ,\text{where} \\
 & C_0:=\ti{P}_1\oplus\ti{P}_0'\ ,\qquad
   C_1:=\tau^h(\ti{P}_0)\oplus\ti{P}_1'\ ,\qquad
   c_0:=\begin{pmatrix} -p_1 & 0 \\ \phi_1 & p_0'\end{pmatrix}\ ,\qquad
   c_1:=\begin{pmatrix} -\tau^h(p_0) & 0 \\ \tau^h(\phi_0) & p_1'\end{pmatrix}\ .
 \end{split}
\end{equation*}
 \label{defn:cone_Z}
\end{defn}
This mapping cone is well-defined. In fact,
one can see that the degree of $c_0$ and $c_1$ are zero and two,
since
the graded ${R}$-homomorphisms $p_1:\ti{P}_1\to\ti{P}_0$ of degree two
and $p_0:\ti{P}_0\to\ti{P}_1$ of degree zero
induce graded ${R}$-homomorphisms $-p_1:\ti{P}_1\to\tau^h(\ti{P}_0)$
of degree zero and $-\tau^h(p_0):\tau^h(\ti{P}_0)\to\ti{P}_1$
of degree two, respectively.

The following is stated in \cite{o:2} Proposition 3.4.
\begin{thm}
The additive category $\tHMF{R}$ endowed
with the shift functor $T$ and the distinguished triangles
forms a triangulated category,
where a distinguished triangle is defined by a sequence
isomorphic to the sequence
\begin{equation*}
 \ti{M}\stackrel{\Phi}{\to}\ti{M}' \to C(\Phi)\to T(\ti{M})\
\end{equation*}
for some $\ti{M},\ti{M}'\in\tMF{R}$
and $\Phi\in\mathrm{Hom}_\tMF{R}(\ti{M},\ti{M}')$.
\end{thm}
\begin{pf}
As in the case for $\HMF{A}$,
the proof is the same as the proof of the analogous
result for a usual homotopic category
(see e.g. \cite{gm:1}, \cite{ks:1}).
\qed\end{pf}
Let $\ti{M}=(\mfs{\ti{P}_0}{p_0}{p_1}{\ti{P}_1})\in\tHMF{R}$
be a graded matrix factorization of size $r$.
One can choose homogeneous free basis
$(b_1,\cdots,b_r;{\bar b}_1,\cdots,{\bar b}_r)$ such that
$\ti{P}_0=b_1{R}\oplus\cdots\oplus b_r{R}$ and
$\ti{P}_1={\bar b}_1{R}\oplus\cdots\oplus{\bar b}_r{R}$.
Then, the graded matrix factorization $\ti{M}$ is expressed
as a pair $(Q,S)$ of $2r$ by $2r$ matrices,
where $S$ is the diagonal matrix of the form
$S:=\mathrm{diag}(s_1,\cdots,s_r; {\bar s}_1,\cdots,{\bar s}_r)$ such that
$s_i=\deg(b_i)$ and ${\bar s}_i=\deg({\bar b}_i)-1$
for $i=1,2,\cdots,r$ and
\begin{equation}\label{Q}
 Q= \begin{pmatrix} {\bf 0} & \varphi \\ \psi & {\bf 0} \end{pmatrix}\ ,\qquad \varphi,\psi\in\mathrm{Mat}_r(R)
\end{equation}
satisfying
\begin{equation}
 Q^2=f\cdot{\bf 1}_{2r}\ ,\qquad -SQ+QS+2EQ= Q\ .
 \label{S}
\end{equation}
We call this $S$ a {\em grading matrix} of $Q$.
This procedure $\ti{M}\mapsto (Q,S)$
gives the equivalence between
the triangulated category $\tHMF{R}$ and
the triangulated category $D^b_{\mathbb Z}({\mathcal A}_f)$ in \cite{t:2}.
This implies that
$\tHMF{R}$ is an enhanced triangulated category
in the sense of Bondal-Kapranov \cite{bk:1}.

We shall represent the matrix factorization
$\ti{M}=(\mfs{\ti{P}_0}{p_0}{p_1}{\ti{P}_1})$
by $(Q,S)$.

\section{$\tHMF{R}$ for type ADE and representations of Dynkin quivers}
\label{sec:main}

In this section, we formulate the main theorem (Theorem \ref{thm:main})
of the present paper
in subsection \ref{ssec:main-state}.
The proof of the theorem is given
in subsection \ref{ssec:proof}.

\subsection{Statement of the main theorem (Theorem \ref{thm:main})}\hfill\\
\label{ssec:main-state}
The main theorem states an equivalence
between the triangulated category
$\tHMF{R}$ for a polynomial $f\in R$ of type ADE
with the derived category of modules over
a {\em path algebra} of a {\em Dynkin quiver}.
In order to formulate the results,
we recall
(i) the weighted homogeneous polynomials of type ADE and
(ii) the notion of the path algebras of the Dynkin quivers.

\vspace*{0.2cm}

\noindent
(i)\ {\bf ADE polynomials.}\ \,
For a regular weight system $W$, we have the following facts
\cite{sa:weight}.

\vspace*{0.15cm}

\noindent
(a)\
There exist integers $m_1,\cdots,m_l$,
called the {\em exponents} of $W$, such that
$\chi_W(T)=T^{m_1}+\cdots + T^{m_l}$, where the smallest
exponent is given by $\epsilon:=a+b+c-h$.

\vspace*{0.15cm}

\noindent
(b)\
The regular weight systems with $\epsilon>0$ are
listed as follows.
\begin{equation}
 \begin{split}
 & A_l : (1,b,l+1-b; l+1)\ ,1\le b\le l\ , \quad
 D_l : (l-2,2,l-1;2(l-1))\ , \\
 & E_6 : (4,3,6;12)\ , \quad
 E_7 : (6,4,9;18)\ , \quad
 E_8 : (10,6,15;30) \ .
 \end{split}
 \label{ADE}
\end{equation}
Here, the naming in the left hand side is given
according to the identifications of the exponents of
the weight systems with those of the simple Lie algebras.
As a consequence,
one obtains $\epsilon=1$ for all regular weight
system with $\epsilon>0$.
For the polynomials of type ADE,
without loss of generality we may choose the followings:
\begin{equation*}
f(x,y,z)=
\left\{\begin{array}{lll}
x^{l+1}+yz,      & h=l+1,    & A_{l}\ ( l\ge 1),\\
x^2y+y^{l-1}+z^2,& h=2(l-1), & D_l\  ( l\ge 4),\\
x^3+y^4+z^2,     & h=12,     & E_6,\\
x^3+xy^3+z^2,    & h=18,     & E_7, \\
x^3+y^5+z^2,     & h=30,     & E_8\ .
\end{array}\right.
\end{equation*}

\vspace*{0.25cm}

\noindent
\ (ii)\ {\bf Path algebras.}\ \,
(a)\ The {\em path algebra} ${\mathbb C}\vec{\Delta}$ of a {\em quiver}
is defined as follows
(see \cite{g:1}, \cite{r:1} and \cite{h} Chapter 1, 5.1).
A {\em quiver} $\vec{\Delta}$ is a pair $(\Delta_0,\Delta_1)$ of
the set $\Delta_0$ of vertices and
the set $\Delta_1$ of arrows (oriented edges).
Any arrow in $\Delta_1$ has a starting point and end point in $\Delta_0$.
A path of {\em length} $r\ge 1$
from a vertex $v$ to a vertex $v'$ in a quiver $\vec{\Delta}$
is of the form $(v|\alpha_1,\cdots,\alpha_r|v')$ with
arrows $\alpha_i\in\Delta_1$ satisfying
the starting point of $\alpha_1$ is $v$,
the end point of $\alpha_i$ is equal to the starting point of
$\alpha_{i+1}$
for all $1\le i\le r-1$,
and the end point of $\alpha_r$ is $v'$.
In addition, we also define
a path of length zero $(v|v)$
for any vertex $v$ in $\vec{\Delta}$.
The {\em path algebra} ${\mathbb C}\vec{\Delta}$ of a quiver $\vec{\Delta}$
is then the ${\mathbb C}$-vector space with basis the set of all paths
in $\vec{\Delta}$.
The product structure is defined by the composition of paths,
where the product of two non-composable paths is set to be zero.

The category of
finitely generated right modules over the path algebra
${\mathbb C}\vec{\Delta}$ is denoted by $\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta}$.
It is an abelian category, and its derived category is
denoted by $D^b(\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta})$.
If $\Delta_0$ and $\Delta_1$ are finite sets
and $\vec{\Delta}$ does not have any
oriented cycle, then ${\mathbb C}\vec{\Delta}$ is a finite dimensional
algebra and $D^b(\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta})$
is a Krull-Schmidt category.

\vspace*{0.15cm}

\noindent
(b)\
A {\em Dynkin quiver} $\vec{\Delta}$ of type ADE
is one of the Dynkin diagram $\Delta$
listed in Figure 1
together with an orientation
for each edge of the diagram.
It is known \cite{g:1} that the number of all the isomorphism classes of
the indecomposable objects of
the abelian category $\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta}$ of a quiver
$\vec{\Delta}$ is finite if and only if the quiver $\vec{\Delta}$ is
a Dynkin quiver (of type ADE).
\begin{equation*}
 \begin{split}
  A_l :\ &\qquad
\xymatrix{
 \bullet_1 \ar@{-}@{-}[r]^{}
  & \bullet_2 \ar@{-}[r]
   & \, \cdots\, \ar@{-}[r]^{}
   & \bullet_b \ar@{-}[r]^{}
   & \, \cdots\cdot\cdot\, \ar@{-}[r]^{}
  & \bullet_{l-1}  \ar@{-}[r]^{}
  & \bullet_l \quad ,
}\\
 D_l:\
&\qquad
\xymatrix{
 & & & & \bullet_{l-1} \\
 \bullet_1 &
 \bullet_2 \ar@{-}[l]^{} \ar@{-}[r]^{} &
 \cdot\cdots\cdot \ar@{-}[r] &
 \bullet_{l-3} \ar@{-}[r] &
 \bullet_{l-2} \ar@{-}[r]^{} \ar@{-}@<1.3ex>[u]^{} &
 \bullet_l \quad ,
}\\
 E_6:\ &\qquad
\xymatrix{
 & & \bullet_1 \ar@{-}@<-0.4ex>[d]^{} & \\
\bullet_5 & \bullet_3 \ar@{-}[l]^{} &
\bullet_2 \ar@{-}[l] \ar@{-}[r]
 & \bullet_4 \ar@{-}[r]^{} & \bullet_6\quad ,
} \\
 E_7 :\ &\qquad
\xymatrix{
 & & \bullet_5 & & \\
 \bullet_7
 & \bullet_6 \ar@{-}[l]
 & \bullet_4 \ar@{-}[l]^{}\ar@{-}@<0.4ex>[u]^{}
 & \bullet_3 \ar@{-}[l]
 & \bullet_2 \ar@{-}[l]^{}
 & \bullet_1 \ar@{-}[l]\quad ,
} \\
 E_8 :\ &\qquad
\xymatrix{
 & & & & \bullet_8 \\
 \bullet_1 \ar@{-}[r]^{}
 & \bullet_2 \ar@{-}[r]
 & \bullet_3 \ar@{-}[r]^{}
 & \bullet_4 \ar@{-}[r]
 & \bullet_5 \ar@{-}[r]^{}\ar@{-}@<0.4ex>[u]^{}
 & \bullet_6 \ar@{-}[r]
 & \bullet_7 \quad .
}
 \end{split}
\end{equation*}
\begin{center}
 F{\scriptsize{IGURE}} 1.\ ADE Dynkin diagram
\end{center}
For a Dynkin diagram $\Delta$,
we shall denote by $\Pi$ the set of vertices.
For later convenience,
the elements of $\Pi$ are labeled by the integers $\{1,\cdots, l\}$
as in the above figures,
and we shall sometimes confuse vertices in $\Pi$ with
labels in $\{1,\cdots, l\}$.

The following is the main theorem of the present paper.
\begin{thm}
Let $f\in{\mathbb C}[x,y,z]$ be a polynomial of type ADE,
and let $\vec{\Delta}$ be a Dynkin quiver
of the corresponding type with a fixed orientation.
Then, we have the following equivalence of the triangulated categories
\begin{equation*}
 \tHMF{R}\simeq D^b(\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta})\ .
\end{equation*}
 \label{thm:main}
\end{thm}

\subsection{The proof of Theorem \ref{thm:main}}\hfill\\
 \label{ssec:proof}
The construction of the proof of Theorem \ref{thm:main} is
as follows.
\begin{itemize}
 \item[Step 1.]\
We describe the Auslander-Reiten (AR-)quiver
for the triangulated category $\HMF{{\hat {\mathcal O}}}$
of matrix factorizations
due to \cite{e:1}, \cite{ar:1} and \cite{a:1} and
give the matrix factorizations explicitly.

 \item[Step 2.]\ We determine the structure of
the triangulated category $\tHMF{R}$ of graded matrix factorizations
(Theorem \ref{thm:grading}).

 \item[Step 3.]\
By comparing the AR-quiver of the
category $\HMF{{\hat {\mathcal O}}}$ with the category $\tHMF{R}$
we find the exceptional collections corresponding to
$\vec{\Delta}$ in $\tHMF{R}$
and complete the proof of the main theorem (Theorem \ref{thm:main}).
\end{itemize}

\noindent
{\bf Step 1.\ \ The Auslander-Reiten quiver for $\HMF{{\hat {\mathcal O}}}$. }

We recall the known results on the
equivalence of the McKay quiver for Kleinean group and
the AR-quiver for the simple singularities \cite{ar:1},
\cite{a:1} and \cite{e:1}.

Let ${\mathcal C}$ be a Krull-Schmidt category over ${\mathbb C}$
which is equivalent to a small category.
An object $X\in\mathrm{Ob}({\mathcal C})$ is called {\em indecomposable}
if any idempotent
$e\in\mathrm{Hom}_{\mathcal C}(X,X)$ is zero or the identity $\mathrm{id}_X$.
For two objects
$X,Y\in\mathrm{Ob}({\mathcal C})$, denote by
$\mathrm{rad}_{\mathcal C}(X,Y)$ the linear subspace of
$\mathrm{Hom}_{\mathcal C}(X,Y)$ of non-invertible morphisms
from $X$ to $Y$.
We denote by $\mathrm{rad}_{\mathcal C}^2(X,Y)\subset\mathrm{rad}_{\mathcal C}(X,Y)$
the space of morphisms each of which is described
as a composition $\Phi'\Phi$ with $\Phi\in\mathrm{rad}_{\mathcal C}(X,Z)$,
$\Phi'\in\mathrm{rad}_{\mathcal C}(Z,Y)$ for some object $Z\in\mathrm{Ob}({\mathcal C})$.
For two indecomposable objects $X,Y\in\mathrm{Ob}({\mathcal C})$,
an element in $\mathrm{rad}_{\mathcal C}(X,Y)\backslash\mathrm{rad}_{\mathcal C}^2(X,Y)$
is called an {\em irreducible morphism}.
The space
$\mathrm{Irr}_{\mathcal C}(X,Y):=\mathrm{rad}_{\mathcal C}(X,Y)/\mathrm{rad}_{\mathcal C}^2(X,Y)$ in fact forms
a subvector space of $\mathrm{Hom}_{\mathcal C}(X,Y)$.
We call
by the {\em AR-quiver $\Gamma({\mathcal C})$} of a Krull-Schmidt category ${\mathcal C}$
the quiver $\Gamma({\mathcal C}):=(\Gamma_0,\Gamma_1)$
whose vertex set $\Gamma_0$ consists of
the isomorphism classes $[X]$ of the indecomposable objects
$X\in\mathrm{Ob}({\mathcal C})$ and
whose arrow set $\Gamma_1$ consists of
$\dim_{\mathbb C}(\mathrm{Irr}_{\mathcal C}(X,Y))$ arrows
from $[X]\in\Gamma_0$ to $[Y]\in\Gamma_0$
for any $[X],[Y]\in\Gamma_0$ (see \cite{r:1},\cite{h},\cite{y}).

On the other hand,
for a Dynkin diagram $\Delta$ listed in Figure 1,
we define a quiver consisting of
the vertex set $\Pi$ and
arrows in both directions
$\mfss{k}{}{}{k}\!'$ for each edge
of $\Delta$ between vertices $k,k'\in\Pi$.
The resulting quiver is denoted by $\overset{\leftrightarrow}{\Delta}$.

Note that the category $\HMF{{\hat {\mathcal O}}}$
is Krull-Schmidt (see \cite{y}, Proposition 1.18).
\begin{thm}
Let $f$ be a polynomial of type ADE, which we regard
as an element of ${\hat {\mathcal O}}$.

{\em (i)}\ $($\cite{ar:1},\cite{a:1},\cite{e:1}$)$
The AR-quiver of the category $\HMF{{\hat {\mathcal O}}}$
is isomorphic to the quiver $\overset{\leftrightarrow}{\Delta}$
corresponding to $f$ $:$
\begin{equation*}
 \Gamma(\HMF{{\hat {\mathcal O}}})\simeq \overset{\leftrightarrow}{\Delta}\ .
\end{equation*}

{\em (ii)}\
According to (i),
fix an identification $\Pi\simeq\Gamma_0(\HMF{{\hat {\mathcal O}}})$,
$k\mapsto [M^k]$.
A representative $M^k$ of the isomorphism classes $[M^k]$
of the indecomposable matrix factorizations
of minimum size is given explicitly in Table \ref{tbl:1}.
The size of $M^k$ is $2\nu_k$,
where $\nu_k$ is the coefficient
of the highest root for $k\in\Pi$.
 \label{thm:AR}
\end{thm}
\begin{pf}
(i)\ This statement follows from the combination of results of
\cite{ar:1} and \cite{e:1},
where the Auslander-Reiten quivers of the categories of
the maximal Cohen-Macaulay modules over ${\hat {\mathcal O}}/(f)$
for type ADE are determined in \cite{ar:1},
and the equivalence of the category of Maximal Cohen-Macaulay modules
with the category $\HMF{{\hat {\mathcal O}}}$ of the matrix factorizations is
given in \cite{e:1}.

(ii)\
Since we have $\sharp(\Pi)$ non-isomorphic
matrix factorizations (Table \ref{tbl:1}),
these actually complete all the vertices of the AR-quiver
$\Gamma(\HMF{{\hat {\mathcal O}}})$.
\qed \end{pf}
\begin{rem}
In \cite{y}, matrix factorizations for a polynomial of type ADE
in two variables $x,y$ are listed up completely.
On the other hand,
for type $A_l$ and $D_l$ in both two and three variables,
all the matrix factorizations and the AR-quivers
are presented in \cite{sch},
where the relation of the results in two variables and those in three
variables is given. This gives a method of finding
the matrix factorizations of a polynomial of type $E_l$, $l=6,7,8$,
in three variables
from the ones in two variables case \cite{y}.
For a recent paper in physics, see also \cite{kl:3}.
 \label{rem:mf}
\end{rem}
Hereafter we fix an identification of
$\Gamma(\HMF{{\hat {\mathcal O}}})$ with $\overset{\leftrightarrow}{\Delta}$
by $k\leftrightarrow [M^k]$.

\vspace*{0.3cm}

\noindent
{\bf Step 2.\ \ Indecomposable objects in $\tHMF{R}$}

Recall that one has the inclusions $R\subset{\mathcal O}\subset{\hat {\mathcal O}}$.
We prepare some definitions
for any fixed weighted homogeneous polynomial $f\in R$.
\begin{defn}[Forgetful functor from $\tHMF{R}$ to $\HMF{{\hat {\mathcal O}}}$]
For a fixed weighted homogeneous polynomial $f\in R$,
there exists a functor $F:\tHMF{R}\to\HMF{{\hat {\mathcal O}}}$
given by $F(\ti{M}):=\ti{M}\otimes_R{\hat {\mathcal O}}$
for $\ti{M}\in\tHMF{R}$ and
the naturally induced homomorphism
$F:\HomDz{R}(\ti{M},\ti{M}')\to\HomD{{\hat {\mathcal O}}}(F(\ti{M}),F(\ti{M}'))$
for any two objects $\ti{M}, \ti{M}'\in\tHMF{R}$.
We call this $F$ the {\em forgetful functor}.
 \label{defn:F}
\end{defn}
It is known that $F:\tHMF{R}\to\HMF{{\hat {\mathcal O}}}$
brings an indecomposable object to
an indecomposable object (\cite{y}, Lemma 15.2.1).

Let us introduce the notion of {\em distance} between
two indecomposable objects in $\HMF{{\hat {\mathcal O}}}$ and in $\tHMF{R}$ as follows.
\begin{defn}[Distance]
For any two indecomposable objects $M,M'\in\HMF{{\hat {\mathcal O}}}$,
define the {\em distance} $d(M,M')\in{\mathbb Z}_{\ge 0}$ from $M$ to $M'$ by
the minimal length of the paths from $[M]$ to $[M']$
in the AR-quiver $\Gamma(\HMF{{\hat {\mathcal O}}})$ of $\HMF{{\hat {\mathcal O}}}$.
In particular, we have $d(M,M')=0$ if and only if $M\simeq M'$
in $\HMF{{\hat {\mathcal O}}}$.

For any two indecomposable objects $\ti{M},\ti{M}'\in\tHMF{R}$,
the {\em distance} $d(\ti{M},\ti{M}')\in{\mathbb Z}_{\ge 0}$
from $\ti{M}$ to $\ti{M}'$ is defined by
\begin{equation*}
 d(\ti{M},\ti{M}'):=d(F(\ti{M}),F(\ti{M}'))\ .
\end{equation*}
 \label{defn:distance}
\end{defn}
By definition,
an irreducible morphism
exists in $\HomD{{\hat {\mathcal O}}}(M,M')$ if and only if $d(M,M')=1$.

Let us return to the case that $f$ is of type ADE.
In this case,
the distance is in fact symmetric:
$d(M^k,M^{k'})=d(M^{k'},M^k)$ for any $[M^k],[M^{k'}]\in\Pi$,
since, due to Theorem \ref{thm:AR}, there exists an arrow from
$[M^k]$ to $[M^{k'}]$ if and only if
there exists an arrow from $[M^{k'}]$ to $[M^k]$.
For two indecomposable objects $\ti{M}^k,\ti{M}^{k'}\in\tHMF{R}$
such that $F(\ti{M}^k)=M^k$, $F(\ti{M}^{k'})=M^{k'}$,
we denote $d(\ti{M}^k,\ti{M}^{k'})=d(M^k,M^{k'}):=d(k,k')$.

In order to state the following Theorem \ref{thm:grading},
it is convenient
to introduce the ordered decomposition $\Pi=\{\Pi_1,\Pi_2\}$
of $\Pi$, called
a {\em principal decomposition} of $\Pi$ \cite{sa:principal},
as follows.
We first define the base vertex $[M_o]\in\Pi$.
For a diagram of $D_l$ or $E_l$, $l=6,7,8$,
we choose $[M_o]$ as the trivalent vertex.
Explicitly, it is $[M^{k=l-2}]$ for type $D_l$,
$[M^{k=2}]$ for type $E_6$,
$[M^{k=4}]$ for type $E_7$ and $[M^{k=5}]$ for type $E_8$ (Figure 1).
For type $A_l$, we set $[M_o]:=[M^{k=b}]$
(see Figure 1) depending on the index $b$, $1\le b\le l$
(see eq.(\ref{ADE})).
Then, we define
the decomposition of $\Pi$ by
\begin{equation*}
 \Pi_1:=\{k\in\Pi\ |\ d(M_o,M^k)\in 2{\mathbb Z}_{\ge 0}+1\}\ ,\qquad
 \Pi_2:=\{k\in\Pi\ |\ d(M_o,M^k)\in 2{\mathbb Z}_{\ge 0}\}\ .
\end{equation*}

Recall that we express a graded matrix factorization
$\ti{M}\in\tHMF{R}$ by the pair $\ti{M}=(Q,S)$,
where $Q$ denotes a matrix factorization in eq.(\ref{Q}) and
$S$ is the grading matrix defined in eq.(\ref{S}).
\begin{thm} Let $f\in R$ be a polynomial of type ADE.
The triangulated category $\tHMF{R}$
is described as follows.

\noindent
{\em (i)}\ {\bf (Objects)}$:$\ \,
The set of isomorphism classes of all indecomposable objects of
$\tHMF{R}$ is given by
\begin{equation*}
[\ti{M}^k_n:=(Q^k,S^k_n)],
\quad k\in\Pi\ ,
\quad n\in{\mathbb Z} \ .
\end{equation*}
Here, $\bullet$\
$Q^k$ is the matrix factorizations of size $2\nu_k$
given in Theorem \ref{thm:AR} (ii) (Table \ref{tbl:1}),

$\bullet$\
the grading matrix $S^k_n$ for $k\in\Pi_\sigma$, $\sigma=1,2$,
and $n\in{\mathbb Z}$ is given by$:$
\begin{equation}
 S^k_n:
=\mathrm{diag}\left(q^k_1,-q^k_1,\cdots,q^k_{\nu_k},-q^k_{\nu_k};
 {\bar q}^k_1,-{\bar q}^k_1,\cdots,{\bar q}^k_{\nu_k},-{\bar q}^k_{\nu_k}\right)
+\phi^k_n\cdot{\bf 1}_{4\nu_k}\ ,
 \label{gradADE}
\end{equation}
where the data of the first term, called the traceless part$:$
$q^k_j\in\frac{2}{h}{\mathbb Z}-\frac{\sigma}{h}$ and
${\bar q}^k_j\in\frac{2}{h}{\mathbb Z}-\frac{\sigma}{h}-1$ for $1\le j\le\nu_k$
are given in {\em Table \ref{tbl:2}}, and
the coefficient of the second term, called the phase,
is given by $\phi^k_n:=\phi(\ti{M}^k_n)=\frac{2n+\sigma}{h}$.

\vspace*{0.1cm}

\noindent
{\em (ii)}\ {\bf (Morphisms)}$:$\ \,
{\em (ii-a)}\ An irreducible morphism exists in
$\HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'})$ if and only if
$d(k,k')=1$ and
$\phi^{k'}_{n'}=\phi^k_n+\ov{h}$.

{\em (ii-b)}\
$\dim_{\mathbb C}(\HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'}))=1$ for
$\phi^{k'}_{n'}-\phi^k_n=\ov{h} d(k,k')$.

\vspace*{0.1cm}

\noindent
{\em (iii)}\ {\bf (The Serre duality)}$:$\ \,
The automorphism ${\mathcal S}:=T\tau^{-1}:\tHMF{R}\to\tHMF{R}$
satisfies the following properties.

{\em (iii-a)}\
$\HomDz{R}(\ti{M}^k_n,{\mathcal S}(\ti{M}^k_n))\simeq{\mathbb C}$
for any indecomposable object $\ti{M}^k_n\in\tHMF{R}$.

{\em (iii-b)}\
This isomorphism of (iii-a) induces the following
bilinear map $:$
\begin{equation*}
 \HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'})
 \otimes\HomDz{R}(\ti{M}^{k'}_{n'},{\mathcal S}(\ti{M}^k_n))\to{\mathbb C}\ ,
\end{equation*}
which is nondegenerate for any $k,k'\in\Pi$ and $n,n'\in{\mathbb Z}$.
 \label{thm:grading}
\end{thm}
\begin{pf}
(i)\
For each $M^k$,
by direct calculations based on the explicit form of $M^k$ in
Table \ref{tbl:1},
we can attach grading matrices $S$ satisfying eq.(\ref{S}).
This, together with the fact that
$F:\tHMF{R}\to\HMF{{\hat {\mathcal O}}}$ brings an indecomposable object
to an indecomposable object, implies that
the union $\coprod_{k=1}^lF^{-1}(M^k)$ gives the set of all
indecomposable objects in $\tHMF{R}$ and then
$F:\tHMF{R}\to\HMF{{\hat {\mathcal O}}}$ is a surjection
(see also \cite{ar:2} and \cite{y}, Theorem 15.14).
Actually, $S$ is unique up to an addition of
a constant multiple of the identity.
Therefore, we decompose $S$ into the traceless part and the phase
part.
Due to the restriction of degrees (\ref{P-grad})
and the definition of $S$,
one has $\pm q_j^k+\phi^k_n\in\frac{2}{h}{\mathbb Z}$ and
$\pm {\bar q}_j^k+\phi^k_n\in\frac{2}{h}{\mathbb Z}-1$ for any
$k\in\Pi$ and $1\le j\le \nu_k$.
By solving these conditions, we obtain
Statement (i), {\it i.e.}, Table \ref{tbl:2}.
In particular, one has
$F^{-1}(M^k)=\{\ti{M}^k_n\ |\ n\in{\mathbb Z}\}$ and
$\tau(\ti{M}^k_n)=\ti{M}^k_{n+1}$.

\vspace*{0.3cm}

\noindent
(ii-a)\
For any two indecomposable objects
$\ti{M}^k_n,\ti{M}^{k'}_{n'}\in\tHMF{R}$,
Statement (i) implies that
$F:\HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'})\to\HomD{{\hat {\mathcal O}}}(M^k,M^{k'})$
is injective and then
$$F:\oplus_{n''\in{\mathbb Z}}\HomDz{R}(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'}))
\simeq\HomD{{\hat {\mathcal O}}}(M^k,M^{k'})$$ is an isomorphism of vector spaces.
This induces the following isomorphisms:
\begin{equation*}
 \begin{array}{cccc}
 F:&\oplus_{n''\in{\mathbb Z}}\mathrm{rad}_{\tHMF{R}}(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'}))
 &\overset{\sim}{\to} &\mathrm{rad}_{\HMF{{\hat {\mathcal O}}}}(M^k,M^{k'}) \ ,\\
     &       \cup        &&  \cup \\
 F:&
 \oplus_{n''\in{\mathbb Z}}\mathrm{rad}_{\tHMF{R}}^2(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'}))
 &\overset{\sim}{\to}&\mathrm{rad}_{\HMF{{\hat {\mathcal O}}}}^2(M^k,M^{k'}) \ ,
 \end{array}
\end{equation*}
and hence, the isomorphism
$F:\oplus_{n''\in{\mathbb Z}}
\mathrm{Irr}_{\tHMF{R}}(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'}))
\overset{\sim}{\to}\mathrm{Irr}_{\HMF{{\hat {\mathcal O}}}}(M^k,M^{k'})$.

For $k,k'\in\Pi$,
define a multi-set $\frak C(k,k')$
of non-negative integers by
\begin{equation*}
\frak C(k,k')
:=\{h(\phi(\tau^{n''}(\ti{M}^{k'}_{n'}))-\phi(\ti{M}^k_n))\ |\
 \HomDz{R}(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'}))\ne 0\ ,\ \ n''\in{\mathbb Z}\}\ ,
\end{equation*}
where we fix $n,n'\in{\mathbb Z}$ for each $F^{-1}(M^k)$ and the integer
$h(\phi(\tau^{n''}(\ti{M}^{k'}_{n'}))-\phi(\ti{M}^k_n))
=h(\phi^{k'}_{n'}-\phi^k_n)+2n''$
appears with multiplicity
$\dim_{\mathbb C}(\HomDz{R}(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'})))$.
The multi-set $\frak C(k,k')$ in fact depends only on
$k$ and $k'$, and is independent of the choice of $n$ and $n'$.

For $k,k'\in\Pi$ such that $d(k,k')=1$,
by calculating
$\HomD{{\hat {\mathcal O}}}(M^k,M^{k'})$
using the explicit forms of the matrix factorizations $M^k, M^{k'}$
in Table \ref{tbl:1},
one can see that $\frak C(k,k')$ consists
of positive odd integers including
$1$ of multiplicity one.
This implies that,
for two indecomposable objects
$\ti{M}^k_n,\ti{M}^{k'}_{n'}\in\tHMF{R}$,
a morphism in $\HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'})$
is an irreducible morphism if and only if
$h(\phi^{k'}_{n'}-\phi^k_n)=1$
(Statement (ii-a)).
(A morphism in $\HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'})$
which corresponds to $c\in\frak C(k,k')$ with $c\ge 3$ can be obtained
by composing irreducible morphisms. )

\vspace*{0.3cm}

\noindent
(ii-b)\
One gets the following lemma:
\begin{lem}\label{lem:A}
{\em 1)}\ For an indecomposable object $\ti{M}^k_n\in\tHMF{R}$, one has
\begin{equation*}
 \text{\em 1-a)}\ \HomDz{R}(\ti{M}^k_n,\ti{M}^k_n)\simeq{\mathbb C}\ ,\qquad
 \text{\em 1-b)}\ \HomDz{R}(\ti{M}^k_n,{\mathcal S}(\ti{M}^k_n))\simeq{\mathbb C}\ .
\end{equation*}
{\em 2)}\ For an indecomposable object $\ti{M}^k_n\in\tHMF{R}$,
a morphism $\Psi\in\HomDz{R}(\ti{M}^k_n,{\mathcal S}(\ti{M}^k_n))$ and
an indecomposable object $\ti{M}^{k'}_{n'}\in\tHMF{R}$ such that
$d(k,k')=1$,

{\em 2-a)}\ $\Psi\Phi\sim 0$ holds for an irreducible morphism
$\Phi\in\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^k_n))$,

{\em 2-b)}\ ${\mathcal S}(\Phi)\Psi\sim 0$ holds for an irreducible morphism
$\Phi\in\HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'})$.
\end{lem}
\begin{pf}
By direct calculations of
$\HomDz{R}(\ti{M}^k_n,\ti{M}^k_n)$ and
$\HomDz{R}(\ti{M}^k_n,{\mathcal S}(\ti{M}^k_n))$
using the explicit forms of
the matrix factorizations in Table \ref{tbl:1} again.
\qed\end{pf}

Note that Lemma \ref{lem:A}.1-a also follows from the AR-quiver
$\Gamma(\HMF{{\hat {\mathcal O}}})$ of $\HMF{{\hat {\mathcal O}}}$ in Theorem \ref{thm:AR}
and Statement (ii-a).

Due to Lemma \ref{lem:A}.1-b,
for any indecomposable object $\ti{M}^k_n\in\tHMF{R}$,
one can consider the cone $C(\Psi)\in\tHMF{R}$ of
a nonzero morphism $\Psi\in\HomDz{R}({\mathcal S}^{-1}(\ti{M}^k_n),\ti{M}^k_n)$,
that is, the cone of a lift of the
morphism $\Psi\in\HomDz{R}({\mathcal S}^{-1}(\ti{M}^k_n),\ti{M}^k_n)$
to a homomorphism in $\mathrm{Hom}_\tMF{R}({\mathcal S}^{-1}(\ti{M}^k_n),\ti{M}^k_n)$.
We denote it simply by $C(\Psi)$, since
two different lifts of the morphism
$\Psi\in\HomDz{R}({\mathcal S}^{-1}(\ti{M}^k_n),\ti{M}^k_n)$ lead
to two cones which are isomorphic to each other in $\tHMF{R}$.

Recall that an {\em Auslander-Reiten (AR-)triangle}
(see \cite{h},\cite{y}, and also an Auslander-Reiten sequence or
equivalently an almost split sequence \cite{ar:ass})
in a Krull-Schmidt triangulated category
${\mathcal C}$ is a distinguished triangle
\begin{equation}\label{ARtri-general}
 X\overset{u}{\to} Y\overset{v}{\to} Z\overset{w}{\to} T(X)
\end{equation}
satisfying the following conditions:
\begin{itemize}
 \item[(AR1)] $X, Z$ are indecomposable objects in $\mathrm{Ob}({\mathcal C})$.
 \item[(AR2)] $w\ne 0$
 \item[(AR3)] If $\Phi:W\to Z$ is not a split epimorphism, then
there exists $\Phi':W\to Y$ such that $v\Phi'=\Phi$.
\end{itemize}
Then, it is known (see \cite{h}, Proposition in 4.3)
that $u$ and $v$ are irreducible morphisms.
\begin{lem}\label{lem:ARtri}
For an indecomposable object $\ti{M}^k_n\in\tHMF{R}$ and
the cone $C(\Psi)\in\tHMF{R}$ of
a nonzero morphism $\Psi\in\HomDz{R}({\mathcal S}^{-1}(\ti{M}^k_n),\ti{M}^k_n)$,
the distinguished triangle
\begin{equation}\label{ARtri}
 \ti{M}^k_n\to C(\Psi)\to\tau(\ti{M}^k_n)\overset{T(\Psi)}{\to}
 T(\ti{M}^k_n)
\end{equation}
is an AR-triangle.
\end{lem}
\begin{pf}
Since by definition the conditions (AR1) and (AR2) are already satisfied,
it is enough to show that the condition (AR3) holds.
Consider the functor $\HomDz{R}(\ti{M}^{k'}_{n'}, - )$
on the distinguished triangle (\ref{ARtri}) for
an indecomposable object $\ti{M}^{k'}_{n'}\in\tHMF{R}$.
One obtains the long exact sequence as follows:
\begin{equation}\label{coh-functor}
 \begin{split}
 & \cdots\to\HomDz{R}(\ti{M}^{k'}_{n'},{\mathcal S}^{-1}(\ti{M}^k_n))
 \overset{\Psi}{\to}\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^k_n) \\
 &\hspace*{3.0cm}\to\HomDz{R}(\ti{M}^{k'}_{n'},C(\Psi))\to \\
 &\qquad\qquad
 \HomDz{R}(\ti{M}^{k'}_{n'},\tau(\ti{M}^k_n))\overset{T(\Psi)}{\to}
 \HomDz{R}(\ti{M}^{k'}_{n'},T(\ti{M}^k_n))
 \to\cdots\ .
 \end{split}
\end{equation}
A nonzero morphism in $\HomDz{R}(\ti{M}^{k'}_{n'},\tau(\ti{M}^k_n))$
is a split epimorphism if and only if
$\ti{M}^{k'}_{n'}=\tau(\ti{M}^k_n)$, in which case
$\HomDz{R}(\ti{M}^{k'}_{n'},\tau(\ti{M}^k_n))\simeq{\mathbb C}$ is spanned
by the identity
$\mathrm{id}_{\ti{M}^{k'}_{n'}}\in\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^{k'}_{n'})$.
If $\ti{M}^{k'}_{n'}\ne\tau(\ti{M}^k_n)$,
due to Lemma \ref{lem:A}.2-a, the map $T(\Psi)$ in the long exact
sequence (\ref{coh-functor})
turns out to be zero,
which implies that we have the surjection
$\HomDz{R}(\ti{M}^{k'}_{n'},C(\Psi))
\to\HomDz{R}(\ti{M}^{k'}_{n'},\tau(\ti{M}^k_n))$ and
then the condition (AR3).
\qed\end{pf}
The existence of the AR-triangle together
with the corresponding long exact sequence (\ref{coh-functor})
leads to the two key lemmas as follows.
\begin{lem}
Let $\ti{M}^k_n\in\tHMF{R}$ be an indecomposable object.
The cone $C(\Psi)$ of a nonzero morphism
$\Psi\in\HomDz{R}({\mathcal S}^{-1}(\ti{M}^k_n),\ti{M}^k_n)$ is
isomorphic to the direct sum of indecomposable objects
$\ti{M}^{k_1}_{n_1}\oplus\cdots\oplus\ti{M}^{k_m}_{n_m}$
for some $m\in{\mathbb Z}_{>0}$
such that $\{k_1,\cdots,k_m\}=\{k'\in\Pi\ |\ d(k,k')=1\}$ and
$\phi(\ti{M}^{k_i}_{n_i})=\phi(\ti{M}^k_n)+\ov{h}$
for any $i=1,\cdots,m$.
 \label{lem:B}
\end{lem}
\begin{pf}
For the AR-triangle in eq.(\ref{ARtri}),
the morphisms
$\ti{M}^k_n\to C(\Psi)$ and
$C(\Psi)\to\tau(\ti{M}^k_n)$ are irreducible and hence
one has $\phi(\ti{M}^{k_i}_{n_i})=\phi(\ti{M}^k_n)+\ov{h}$
and $d(k',k_i)=1$,
for any $i=1,\cdots, m$, with the direct sum decomposition of
indecomposable objects
$C(\Psi)\simeq\oplus_{i=1}^m\ti{M}^{k_i}_{n_i}$ above.
Also, the fact that the distinguished triangle (\ref{ARtri})
is an AR-triangle further guarantees that,
for an indecomposable object $\ti{M}^{k'}_{n'}\in\tHMF{R}$,
there exists an irreducible morphism in
$\HomDz{R}(\ti{M}^{k}_{n},\ti{M}^{k'}_{n'})$
if and only if $(k',n')=(k_i,n_i)$ for some $i=1,\cdots, m$
(see Lemma in p.40 of \cite{h}).
Thus, the rest of the proof is to show $k_i\ne k_j$ if $i\ne j$,
for which it is enough to check this lemma
at the level of the grading matrices
(see eq.(\ref{cone-grading}) below Table \ref{tbl:2}).
\qed\end{pf}
\begin{lem}
For an indecomposable object $\ti{M}^{k}_{n}\in\tHMF{R}$,
let $C(\Psi)$ be the cone of a nonzero morphism
$\Psi\in\HomDz{R}({\mathcal S}^{-1}(\ti{M}^{k}_{n}),\ti{M}^{k}_{n})$,
and let
$C(\Psi)\simeq\oplus_{i=1}^m\ti{M}^{k_i}_{n_i}$,
$m\in{\mathbb Z}_{>0}$,
be the direct sum decomposition of indecomposable objects as above.
Then, for an indecomposable object
$\ti{M}^{k'}_{n'}\in\tHMF{R}$ and
the given multi-set $\frak C(k',k)$, one obtains
\begin{equation*}
 \coprod_{i=1}^m\frak C(k',k_i)
 =\{c-1\ |\ c\in\frak C(k',k), c\ne 0\}
 \coprod
 \{c+1\ |\ c\in\frak C(k',k), c\ne h-2\ \text{if}\ k={k'}^{\mathcal S}\ \}\ ,
\end{equation*}
where, for $k\in\Pi$,
$k^{\mathcal S}\in\Pi$ denotes the vertex such that
$[F({\mathcal S}(\ti{M}^k_n))]=[M^{k^{\mathcal S}}]\in\Pi$ for any $n\in{\mathbb Z}$.
 \label{lem:C}
\end{lem}
\begin{pf}
Consider the long exact sequence (\ref{coh-functor}) for the
AR-triangle (\ref{ARtri}).
As discussed in the proof of Lemma \ref{lem:ARtri},
if $\ti{M}^{k'}_{n'}\ne\tau(\ti{M}^k_n)$,
the map $T(\Psi)$ in eq.(\ref{coh-functor})
is zero due to Lemma \ref{lem:A}.2-a.
Similarly, if $\ti{M}^{k'}_{n'}\ne{\mathcal S}^{-1}(\ti{M}^k_n)$,
the map $\Psi$ in eq.(\ref{coh-functor})
is zero due to Lemma \ref{lem:A}.2-b.
On the other hand, if $\ti{M}^{k'}_{n'}=\tau(\ti{M}^k_n)$,
the map $T(\Psi)$ in eq.(\ref{coh-functor}) is
an isomorphism and hence the map
$\HomDz{R}(\ti{M}^{k'}_{n'},C(\Psi))
\to\HomDz{R}(\ti{M}^{k'}_{n'},\tau(\ti{M}^k_n))$ turns out to be zero.
Similarly, if $\ti{M}^{k'}_{n'}={\mathcal S}^{-1}(\ti{M}^k_n)$,
the map $\Psi$ in eq.(\ref{coh-functor}) is
an isomorphism and hence the map
$\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^k_n)
\to\HomDz{R}(\ti{M}^{k'}_{n'},C(\Psi))$
turns out to be zero.
Combining these facts,
one has the exact sequences as follows:
\begin{itemize}
 \item $0\to\mathrm{Hom}(\ti{M}^k_n,\ti{M}^{k'}_{n'})
\to\HomDz{R}(\ti{M}^k_n,C(\Psi))
\to\mathrm{Hom}(\ti{M}^k_n,\tau(\ti{M}^{k'}_{n'}))\to 0$
if $\ti{M}^k_n\ne\tau(\ti{M}^{k'}_{n'})$ and
$\ti{M}^k_n\ne{\mathcal S}^{-1}(\ti{M}^{k'}_{n'})$,

 \item
 $0\to\mathrm{Hom}(\ti{M}^k_n,\ti{M}^{k'}_{n'})
 \to\HomDz{R}(\ti{M}^k_n,C(\Psi))\to 0$
if $\ti{M}^k_n=\tau(\ti{M}^{k'}_{n'})$, and

 \item
 $0\to\HomDz{R}(\ti{M}^k_n,C(\Psi))
\to\mathrm{Hom}(\ti{M}^k_n,\tau(\ti{M}^{k'}_{n'}))\to 0$
if $\ti{M}^k_n={\mathcal S}^{-1}(\ti{M}^{k'}_{n'})$, though in this case
$\mathrm{Hom}(\ti{M}^k_n,\tau(\ti{M}^{k'}_{n'}))=0$.
\end{itemize}
{}From these results follows the statement of this lemma.
\qed\end{pf}
Calculate $\HomDz{R}(M^k,\tau^n(M^k))$, $n\in{\mathbb Z}$,
where we set $k=1$ for type $A_l$, $D_l$ and $E_8$, $k=5$ or $6$ for $E_6$,
and $k=7$ for $E_7$,
by using the explicit forms of matrix factorizations $M^k$
in Table \ref{tbl:1}.
Then, one obtains $\frak C(k,k)$.
Using Lemma \ref{lem:B} and Lemma \ref{lem:C} repeatedly,
one can actually obtain $\frak C(k',k'')$ for all $k',k''\in\Pi$
(Table \ref{tbl:3}).
In particular, by a use of Table \ref{tbl:3},
one can get Statement (ii-b) and
\begin{cor}
The following equivalent statements hold: for two indecomposable
objects $\ti{M}^k_n,\ti{M}^{k'}_{n'}\in\tHMF{R}$,

{\em (a)}\ If
$\phi({\mathcal S}(\ti{M}^{k'}_{n'}))-\phi(\ti{M}^{k}_{n})<
\ov{h}d(k,{k'}^{\mathcal S})$,
then $\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^{k}_{n})=0$.

{\em (b)}\ If
$\phi(\ti{M}^{k}_{n})-\phi(\ti{M}^{k'}_{n'})
<\ov{h}d(k',k)$,
then $\HomDz{R}(\ti{M}^{k}_{n},{\mathcal S}(\ti{M}^{k'}_{n'}))=0$.
 \label{cor:serre1}
\qed\end{cor}

\vspace*{0.3cm}

\noindent
(iii-a)\
This is already proven in Lemma \ref{lem:A}.1-b.

\vspace*{0.3cm}

\noindent
(iii-b)\
Suppose that there exists a nonzero morphism
$\Phi\in\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^{k}_{n})$
for two indecomposable objects
$\ti{M}^k_n,\ti{M}^{k'}_{n'}\in\tHMF{R}$ and
let us show that there exists a nonzero morphism
$\Phi^{\mathcal S}\in\HomDz{R}(\ti{M}^{k}_{n},{\mathcal S}(\ti{M}^{k'}_{n'}))$
such that $\Phi^{\mathcal S}\Phi$ is nonzero in
$\HomDz{R}(\ti{M}^{k'}_{n'},{\mathcal S}(\ti{M}^{k'}_{n'}))$.
If $\ti{M}^{k}_{n}={\mathcal S}(\ti{M}^{k'}_{n'})$, then
one may take $\Phi^{\mathcal S}=\mathrm{id}_{{\mathcal S}(\ti{M}^{k'}_{n'})}$,
so assume that $\ti{M}^{k}_{n}\ne{\mathcal S}(\ti{M}^{k'}_{n'})$,
{\it i.e.}, $0\le \phi^{k}_{n}-\phi^{k'}_{n'}< 1-\frac{2}{h}$.
As in the proof of Lemma \ref{lem:B},
from the long exact sequence (\ref{coh-functor}) of the
AR-triangle (\ref{ARtri}) one obtains the exact sequence
\begin{equation*}
0\to\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^{k}_{n})
\to\HomDz{R}(\ti{M}^{k'}_{n'},C(\Psi))
\end{equation*}
for the cone $C(\Psi)$
of a nonzero morphism $\Psi:{\mathcal S}^{-1}(\ti{M}^{k}_{n})\to\ti{M}^{k}_{n}$.
Namely, there exists an indecomposable object
$\ti{M}^{k''}_{n''}\in\tHMF{R}$
such that $d(k,k'')=1$,
$\phi^{k''}_{n''}-\phi^{k}_{n}=\ov{h}$ and
$\Phi'\Phi$ is not zero in
$\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^{k''}_{n''})$
with an irreducible morphism
$\Phi'\in\HomDz{R}(\ti{M}^{k}_{n},\ti{M}^{k''}_{n''})$.
Repeating this procedure
together with Corollary \ref{cor:serre1} (a) leads that
the path corresponding to the morphism arrives at
${\mathcal S}(\ti{M}^{k'}_{n'})$.
Namely, for a given morphism $\Phi$ in
$\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^{k}_{n})$,
there exists a nonzero morphism
$\Phi^{\mathcal S}\in\HomDz{R}(\ti{M}^{k}_{n},{\mathcal S}(\ti{M}^{k'}_{n'}))$
such that $\Phi^{\mathcal S}\Phi$ is nonzero in
$\HomDz{R}(\ti{M}^{k'}_{n'},{\mathcal S}(\ti{M}^{k'}_{n'}))$.

Conversely, suppose that there exists a morphism
$\Phi^{\mathcal S}\in\HomDz{R}(\ti{M}^{k}_{n},{\mathcal S}(\ti{M}^{k'}_{n'}))$
for two indecomposable objects
$\ti{M}^{k}_{n},\ti{M}^{k'}_{n'}\in\tHMF{R}$.
One can show that there exists a nonzero morphism
$\Phi\in\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^{k}_{n})$
such that $\Phi^{\mathcal S}\Phi$ is nonzero in
$\HomDz{R}(\ti{M}^{k'}_{n'},{\mathcal S}(\ti{M}^{k'}_{n'}))$ by
considering the functor $\HomDz{R}(-,{\mathcal S}(\ti{M}^{k'}_{n'}))$
on the AR-triangle (\ref{ARtri})
together with Corollary \ref{cor:serre1} (b),
instead of $\HomDz{R}(\ti{M}^{k'}_{n'},-)$
with Corollary \ref{cor:serre1} (a).
Thus,
one gets Statement (iii-b) of Theorem \ref{thm:grading}.
\qed\end{pf}

\vspace*{0.3cm}

\noindent
{\bf Step 3.\ \ Exceptional collections in $\tHMF{R}$}

In this step, for a fixed polynomial $f$ of type ADE
and a Dynkin quiver $\vec{\Delta}$
of the corresponding type,
we find an exceptional collection in $\tHMF{R}$
and then show the equivalence of the category $\tHMF{R}$ with
the derived category of modules over
the path algebra of $\vec{\Delta}$.
\begin{lem}
For any Dynkin quiver $\vec{\Delta}$,
there exists ${\bf n}:=(n_1,\cdots,n_l)\in{\mathbb Z}^l$ such that
one has an isomorphism of ${\mathbb C}$-algebras$:$
\begin{equation*}
{\mathbb C}\vec{\Delta}\simeq
{\C\vec{\Gamma}(\bn)}:=\bigoplus_{k,k'\in\Pi}
\HomDz{R}(\ti{M}^k_{n_k},\ti{M}^{k'}_{n_{k'}})\ ,
\end{equation*}
with the natural correspondence between
the path from $k\in\Pi$ to $k'\in\Pi$ and
the space $\mathrm{Hom}(\ti{M}^k_{n_k},\ti{M}^{k'}_{n_{k'}})$
for any $k,k'\in\Pi$.
Such a tuple of integers is unique up to the ${\mathbb Z}$ shift
$(n_1+n,\cdots,n_l+n)$ for some $n\in{\mathbb Z}$.
 \label{lem:Calg}
\end{lem}
\begin{pf}
Let us fix $n_1\in{\mathbb Z}$ of $\ti{M}^1_{n_1}$.
For each $k\in\Pi$, $d(1,k)=1$,
take $\ti{M}^k_{n_k}\in\tHMF{R}$
such that $\phi^k_{n_k}-\phi^1_{n_1}=\ov{h}$ (resp. $-\ov{h}$)
if there exists an arrow in $\vec{\Delta}$ from $1\in\Pi$ to $k\in\Pi$
(resp. from $k\in\Pi$ to $1\in\Pi$).
Repeating this process leads to the tuple
${\bf n}=(n_1,\cdots,n_l)$
such that
${\C\vec{\Gamma}(\bn)}\simeq{\mathbb C}\vec{\Delta}$.
On the other hand,
Theorem \ref{thm:grading} (ii-b) implies that
there exists
a morphism for any given path and there are no relations
among them.
Thus, one gets this lemma.
\qed\end{pf}
\begin{cor}
Let ${\frak S}_l$ be the permutation group of\, $\Pi=\{1,\cdots,l\}$.
Given a Dynkin quiver $\vec{\Delta}$ and
${\bf n}\in{\mathbb Z}^l$ such that
${\C\vec{\Gamma}(\bn)}\simeq{\mathbb C}\vec{\Delta}$,
one can take an element $\sigma\in{\frak S}_l$ such that
$\phi^{\sigma(k')}_{n_{\sigma(k')}}\ge
\phi^{\sigma(k)}_{n_{\sigma(k)}}$ if
$k'> k$. Thus,
for
$\{E^1,\cdots,E^l\}$, $E^k:=\ti{M}^{\sigma(k)}_{n_{\sigma(k)}}$,
$\HomDz{R}(E^k, E^{k'})\ne 0$ only if $k'>k$.
 \label{cor:E}
\qed\end{cor}
\begin{lem}
Given a Dynkin quiver $\vec{\Delta}$ and
${\bf n}\in{\mathbb Z}^l$ such that ${\C\vec{\Gamma}(\bn)}\simeq{\mathbb C}\vec{\Delta}$,
one has $\HomDz{R}(\tau^n(\ti{M}^k_{n_k}),\ti{M}^{k'}_{n_{k'}})=0$
for any $k,k'\in\Pi$ if $n\ge 1$.
 \label{lem:hom-bound}
\qed\end{lem}
\begin{pf}
Let us first concentrate on irreducible morphisms.
For any $k,k'\in\Pi$ such that $d(k,k')=1$,
one obtains
$\phi^{k'}_{n_{k'}}-\phi^k_{n_k+n}=\frac{\pm 1-2n}{h}$
since $\phi^{k'}_{n_{k'}}-\phi^k_{n_k}=\pm\ov{h}$.
On the other hand,
Theorem \ref{thm:grading} (ii-a) implies that there exists an irreducible
morphism in $\HomDz{R}(\ti{M}^k_{n_k+n},\ti{M}^{k'}_{n_{k'}})$
only if $d(k,k')=1$ and $n=0$ or $n=-1$,
since $\frac{\pm 1-2n}{h}$ can be $\ov{h}$ only if $n=0$ or $n=-1$.

Since, by definition, all morphisms except the identities
can be described by compositions of irreducible morphisms,
one gets this lemma.
\qed\end{pf}
By Lemma \ref{lem:hom-bound} and
the Serre duality (Theorem \ref{thm:grading} (iii)),
we obtain the following lemma.
\begin{lem}
Given a Dynkin quiver $\vec{\Delta}$ and ${\bf n}\in{\mathbb Z}^l$ such that
${\C\vec{\Gamma}(\bn)}\simeq{\mathbb C}\vec{\Delta}$,
$\{E^1,\dots, E^l\}$ given in Corollary \ref{cor:E}
is a strongly exceptional collection, that is,
\begin{equation*}
 \begin{cases}
 \HomDz{R}(E^i,E^j)=0 & \text{for}\ \, i>j \ ,\\
 \HomDz{R}(E^i,T^k(E^j))=0  &\text{for}\ \, k\ne 0\ \text{and any}\ \,i,j\ .
 \end{cases}
\end{equation*}
 \label{lem:SEC}
\end{lem}
\begin{pf}
Due to Corollary \ref{cor:E},
it is sufficient to show that
$\HomDz{R}(E^i,T^n(E^j))=0$ for $n\ge 1$.
On the other hand, by the Serre duality,
\begin{equation*}
\dim_{\mathbb C}(\HomDz{R}(E^i,T^n(E^j)))
 =\dim_{\mathbb C}(\HomDz{R}(T^{n-1}\tau(E^j),E^i))
\end{equation*}
holds. If $n=1$,
$\HomDz{R}(T^{n-1}\tau(E^j),E^i)=\HomDz{R}(\tau(E^j),E^i)=0$
follows from Lemma \ref{lem:hom-bound}.
If $n\ge 2$,
first we have
$\phi(T^{n-1}\tau(E^j))=\phi(E^j)+(n-1+\frac{2}{h})$ and then
\begin{equation*}
 \phi(E^i)-\phi(T^{n-1}\tau(E^j))=(\phi(E^i)-\phi(E^j))
 -\left(n-1+\frac{2}{h}\right)\ .
\end{equation*}
Here, it is clear that
\begin{equation*}
 |\phi(E^i)-\phi(E^j)|\le \frac{l-1}{h}
\end{equation*}
holds for $\{E^1,\cdots,E^l\}$.
Thus, we have the following inequality:
\begin{equation*}
 \phi(E^i)-\phi(T^{n-1}\tau(E^j))
\le \frac{l-1}{h}-\left(n-1+\frac{2}{h}\right)< 0\quad \text{if}\ n\ge 2\ .
\end{equation*}
On the other hand,
since all morphisms except the identities
can be obtained by the composition of
the irreducible morphisms, Theorem \ref{thm:grading} (ii-a)
implies that
$\HomDz{R}(E^j,E^i)\ne 0$ only if
$\phi(E^i)-\phi(E^j)\ge\ov{h}d(E^j,E^i)\ge 0$.
This implies \\ $\HomDz{R}(T^{n-1}\tau(E^j),E^i)=0$.
\qed\end{pf}
\begin{cor}
$D^b(\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta})$ is a full triangulated
subcategory of $\tHMF{R}$.
\end{cor}
\begin{pf}
Use the fact that $\{E^1,\dots,E^l\}$ is a strongly exceptional
collection and
$$
{\mathbb C}\vec{\Delta}\simeq \bigoplus_{i,j=1}^{l}\HomDz{R}(E^i,E^j)\ .
$$
Since $\tHMF{R}$ is an enhanced triangulated category,
we can apply the theorem by Bondal-Kapranov $($\cite{bk:1} Theorem
1$)$ which implies that $D^b(\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta})$ is
full as a triangulated subcategory.
\qed\end{pf}
Recall that the number of indecomposable objects of $\tHMF{R}$
up to the shift functor $T$
is $\frac{l\cdot h}{2}$,
which coincides with the number of the positive roots for
the root system of type ADE.
This number is the number of indecomposable objects
of $D^b(\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta})$
up to the shift functor $T$
by Gabriel's theorem \cite{g:1}.
Therefore, one obtains $D^b(\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta})\simeq \tHMF{R}$.

\section{A stability condition on $\tHMF{R}$}
\label{sec:stab}
Let $K_0(\tHMF{R})$ be the Grothendieck group
of the category $\tHMF{R}$ \cite{gr} (see also \cite{h}).
For a triangulated category ${\mathcal C}$, let $F$ be
a free abelian group generated by
the isomorphism classes of objects $[X]$ for $X\in\mathrm{Ob}({\mathcal C})$.
Let $F_0$ be the subgroup generated by
$[X]-[Y]+[Z]$ for all  distinguished triangles
$X\to Y\to Z\to T(X)$ in ${\mathcal C}$.
The Grothendieck group $K_0({\mathcal C})$
of the triangulated category ${\mathcal C}$ is, by definition,
the factor group $F/F_0$.
\begin{defn}\label{defn:centralcharge}
For a graded matrix factorization $\ti{M}=(Q,S)\in\tHMF{R}$,
we associate a complex number as follows:
\begin{equation*}
{\mathcal Z}(\ti{M}):= \mathrm{Tr}(e^{\pi\sqrt{-1}S})\ .
\end{equation*}
\end{defn}
The map ${\mathcal Z}$ extends to a group homomorphism
${\mathcal Z}: K_0(\tHMF{R})\to{\mathbb C}$.
\begin{thm}\label{thm:stab}
Let $f\in R$ be a polynomial of type ADE.
Let ${\mathcal P}(\phi)$ be the full additive subcategory of
$\tHMF{R}$
additively generated by the indecomposable objects of phase $\phi$.
Then ${\mathcal P}(\phi)$ and ${\mathcal Z}$ define a stability condition on
$\tHMF{R}$ in the sense of Bridgeland \cite{bd:1}.
\end{thm}
\begin{pf}
By definition, what we should check is that
${\mathcal P}(\phi)$ and ${\mathcal Z}$ satisfy the following properties$:$
\begin{enumerate}
\item if $\ti{M}\in {\mathcal P}(\phi)$, then
${\mathcal Z}(\ti{M})=m(\ti{M})\exp(\pi\sqrt{-1}\phi)$
for some $m(\ti{M})\in{\mathbb R}_{> 0}$,
\item for all $\phi\in{\mathbb R}$, ${\mathcal P}(\phi+1)=T({\mathcal P}(\phi))$,
\item if $\phi_1>\phi_2$ and $\ti{M}_i\in
{\mathcal P}(\phi_i)$, $i=1,2$, then
$\HomDz{R}(\ti{M}_1,\ti{M}_2)=0$,
\item for each nonzero object $\ti{M}\in \tHMF{R}$,
there is a finite sequence of real numbers
$$
\phi_1 >\phi_2>\dots >\phi_n
$$
and a collection of distinguished triangles
\[
\xymatrix@C=.5em{ \underset{}{\overset{}{0}}
 \ar@{=}[r] & \underset{}{\overset{}{\ti{M}_0}}
 \ar[rr] && \underset{}{\overset{}{\ti{M}_1}} \ar[rr]
\ar[dl] && \underset{}{\overset{}{\ti{M}_2}}
 \ar[rr] \ar[dl] &&\ \ \ldots\ldots\ \
 \ar[rr] && \underset{}{\overset{}{\ti{M}_{n-1}}}
 \ar[rr] && \underset{}{\overset{}{\ti{M}_n}}
 \ar[dl] \ar@{=}[r] &  \underset{}{\overset{}{\ti{M}}} \\
&& \ti{N}_1 \ar@{-->}[ul] && \ti{N}_2 \ar@{-->}[ul] &&&&&& \ti{N}_n
\ar@{-->}[ul] }
\]
with $\ti{N}_j\in {\mathcal P}(\phi_j)$ for all $j=1,\cdots, n$.
\end{enumerate}
Here, for any indecomposable object $\ti{M}^k_n\in\tHMF{R}$,
${\mathcal Z}(\ti{M}^k_n)$ is of the form:
\begin{equation*}
 {\mathcal Z}(\ti{M}^k_n) =m(\ti{M}^k_n)e^{\pi\sqrt{-1}\phi^k_n}\ ,\qquad
 m(\ti{M}^k_n)\in{\mathbb R}_{> 0}\ .
\end{equation*}
In fact, by a straightforward calculation, one obtains
\begin{equation*}
 \begin{split}
 {\mathcal Z}(\ti{M}^k_n)
 & =\sum_{j=1}^{\nu_k}
 \left(e^{\pi\sqrt{-1}(q^k_j+\phi^k_n)}+
 e^{\pi\sqrt{-1}(-q^k_j+\phi^k_n)}+
 e^{\pi\sqrt{-1}({\bar q}^k_j+\phi^k_n)}+
 e^{\pi\sqrt{-1}(-{\bar q}^k_j+\phi^k_n)}
 \right) \\
 &=2 e^{\pi\sqrt{-1}\phi^k_n}\sum_{j=1}^{\nu_k}
 \left(\cos({\pi q^k_j})+  \cos({\pi{\bar q}^k_j})\right)\\
 &=m(\ti{M}^k_n)e^{\pi\sqrt{-1}\phi^k_n}\ ,\qquad
 m(\ti{M}^k_n):=
 2\sum_{j=1}^{\nu_k}\left(\cos({\pi q^k_j})+  \cos({\pi{\bar q}^k_j})\right)\ .
 \end{split}
\end{equation*}
This shows Statement (i) in Theorem \ref{thm:stab}.
It is clear that
Statement (ii),(iii) and (iv) follow from Theorem \ref{thm:grading}.
\qed\end{pf}

Proposition 5.3 in \cite{bd:1} states that
a stability condition $({\mathcal Z},{\mathcal P})$
on a triangulated category defines an abelian category.
In our case, for the triangulate category $\tHMF{R}$,
we obtain an abelian category ${\mathcal P}((0,1])$
which consists of objects $\ti{M}\in\tHMF{R}$
having the decomposition $\ti{N}_1,\cdots,\ti{N}_n$ given by
Theorem \ref{thm:stab} (iv) such that
$1\ge\phi_1>\cdots>\phi_n>0$.
\begin{prop}\label{prop:abelian}
Given a polynomial $f$ of type ADE,
the following equivalence of abelian categories holds:
\begin{equation*}
  {\mathcal P}((0,1])\simeq \mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta}_{principal}\ ,
\end{equation*}
where $\vec{\Delta}_{principal}$ is the Dynkin quiver of the
corresponding ADE Dynkin diagram $\Delta$ with the
principal orientation \cite{sa:principal}, {\it i.e.},
all arrows start from $\Pi_1$ and end on $\Pi_2$.
\qed\end{prop}
\begin{pf}
Let us take the following collection
$\{\ti{M}^1_0,\cdots,\ti{M}^l_0\}$. The corresponding ${\mathbb C}$-algebra
is denoted by ${\mathbb C}\vec{\Gamma}({\bf n}=0)$ (see Lemma \ref{lem:Calg}).
First, we show that
\begin{equation}\label{Calg-principal}
 {\mathbb C}\vec{\Gamma}({\bf n}=0)\simeq{\mathbb C}\vec{\Delta}_{principal}\ ,
\end{equation}
for which
it is enough to say that $\ti{M}^k_0\in{\mathcal P}((0,1])$ is projective
for each $k\in\Pi$.
For any $\ti{N}\in{\mathcal P}(\phi(\ti{N}))$ such that $0<\phi(\ti{N})\le 1$,
the Serre duality implies
\begin{equation*}
 \HomDz{R}(\ti{M}^k_0,T(\ti{N}))
 \simeq \HomDz{R}(\tau(\ti{N}),\ti{M}^k_0)\ .
\end{equation*}
Here, one has $\HomDz{R}(\tau(\ti{N}),\ti{M}^k_0)\ne 0$ only if
$\phi(\ti{N})+\frac{2}{h}\le\phi(\ti{M}^k_0)$.
However, such $\ti{N}$ does not exist since
$\phi(\ti{M}^k_0)=\frac{\sigma}{h}$ for $k\in\Pi_\sigma$, $\sigma=1,2$.
Thus, one has $\HomDz{R}(\ti{M}^k_0,T(\ti{N}))=0$ and hence
eq.(\ref{Calg-principal}).

On the other hand, the full abelian subcategory
$\langle\ti{M}^1_0,\cdots,\ti{M}^l_0\rangle$ of ${\mathcal P}((0,1])$
generated by $\ti{M}^1_0,\cdots,\ti{M}^l_0$
is equivalent to the abelian category
$\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta}_{principal}$.
Also, the Gabriel's theorem \cite{g:1} asserts that
the number of the indecomposable objects
in $\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta}_{principal}$
is equal to the number of the positive roots of $\Delta$,
which coincides with the number of the indecomposable
objects in ${\mathcal P}((0,1])$.
Thus, one obtains the equivalence
$\mathrm{mod}\,\text{-}\,{\mathbb C}\vec{\Delta}_{principal}\simeq{\mathcal P}((0,1])$.
\qed\end{pf}
\begin{rem}[Principal orientation]
In the triangulated category $\tHMF{R}$,
the principal oriented quiver $\vec{\Delta}_{principal}$
is realized by a strongly exceptional collection
$\{\ti{M}^1_{n_1},\cdots,\ti{M}^l_{n_l}\}$ with
$n_1=\cdots =n_l=n\in{\mathbb Z}$ for some $n\in{\mathbb Z}$ as above.
It is interesting that
a strongly exceptional collection of this type
has minimum range of the phase:
$\frac{2n+1}{h}\le\phi(\ti{M}^k_{n_k})\le\frac{2n+2}{h}$
for any $k\in\Pi$.
 \label{rem:principal}
\end{rem}

 \section{Tables of data for matrix factorizations of type ADE}
\label{sec:tbl}

\begin{tbl}
We list up the Auslander-Reiten (AR)-quiver
of the category $\HMF{{\hat {\mathcal O}}}$ for a polynomial
$f$ of type ADE.
We label each isomorphism class of the indecomposable objects
(matrix factorizations) in $\MF{{\hat {\mathcal O}}}$
by upperscript $\{1,\cdot\cdot,k,\cdot\cdot,l\}$ such as $[M^k]$.
We also list up a representative $M^k$ of $[M^k]$
by giving the pair $(\varphi^k,\psi^k)$ for each
matrix factorization $Q^k:=\left(\begin{smallmatrix}  & \psi^k\\ \varphi^k & \end{smallmatrix}\right)$
(see eq.(\ref{Q})).
In addition, for type $D_l$ and $E_l\ (l=6,7,8)$ cases,
we attach the indices
$\left(\begin{smallmatrix} q^k_1\\ \vdots \\ q^k_{\nu_k}\end{smallmatrix}\right)$
defined in Theorem \ref{thm:grading} (eq.(\ref{gradADE}))
to each of those pairs as
\begin{equation*}
 \varphi^k_{\left(\begin{smallmatrix} q^k_1\\ \vdots \\ q^k_{\nu_k}\end{smallmatrix}\right)},
\psi^k_{\left(\begin{smallmatrix} q^k_1\\ \vdots \\ q^k_{\nu_k}\end{smallmatrix}\right)} \ .
\end{equation*}
Those indices are for later convenience and
irrelevant for the statement of Theorem \ref{thm:AR}.
They will be listed up again in the next table
(Table \ref{tbl:2}).

\noindent
{\bf Type $A_l$.}\ \, The AR-quiver for the polynomial
$f=x^{l+1}+yz$ is of the form:
\begin{equation*}
\xymatrix{
 \ \ [M^1]\ \ar@<0.5ex>[r]^{}
  & \ [M^2]\ \ar@<0.5ex>[l]^{}\ar@<0.5ex>[r]^{}
  & \ \cdot\cdots\cdot\ \ar@<0.5ex>[l]^{}\ar@<0.5ex>[r]^{}
  & \ [M^{l-1}]\ \ar@<0.5ex>[l]^{}\ar@<0.5ex>[r]^{}
  & \ [M^l]\ \ar@<0.5ex>[l]^{}\ \ ,
}
\end{equation*}
where, for each $k\in\Pi$, $(\varphi^k,\psi^k)$ is given by
{\small
\begin{equation*}
\varphi^k:=
\begin{pmatrix}
y & x^{l+1-k}\\
x^k & -z
\end{pmatrix},\quad \psi^k:=
\begin{pmatrix}
z & x^{l+1-k}\\
x^k & -y
\end{pmatrix}\ .
\end{equation*}
}
\vspace*{0.3cm}

\noindent
{\bf Type $D_l$.}\ \, The AR-quiver for the polynomial
$f=x^2y+y^{l-1}+z^2$ is given by:
\begin{equation*}
\xymatrix@C=.6em{
 && && && && &\ [M^{l-1}]\  \ar@<0.5ex>[ld]^{}\\
\ [M^1]\ \ar@<0.5ex>[rr]^{}
  && \ [M^2]\ \ar@<0.5ex>[ll]^{}
 \ar@<0.5ex>[rr]^{}
  && \ \  {\cdots} \ \  \ar@<0.5ex>[ll]^{}
  \cdots \ar@<0.5ex>[rr]^{}
  &&\ [M^{l-3}]\
 \ar@<0.5ex>[rr]^{} \ar@<0.5ex>[ll]^{}
  &&\ [M^{l-2}]\
 \ar@<0.5ex>[ru]^{} \ar@<0.5ex>[rd]^{}
 \ar@<0.5ex>[ll]^{} \\
 && && && && &
 \ \ \ [M^l]\ \ar@<0.5ex>[lu]^{}  \ \ ,
}
\end{equation*}
where, for each
$k\in\Pi$, $(\varphi^k,\psi^k)$ is given by
{\small
\begin{equation*}
  \varphi^1_{\left(\begin{smallmatrix} l-3\end{smallmatrix}\right)}=
  \psi^1_{\left(\begin{smallmatrix} l-3\end{smallmatrix}\right)}=
\begin{pmatrix} z & x^2+y^{l-2}\\
    y & -z \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
  \varphi^k_{\left(\begin{smallmatrix} l-2-k\\l-k \end{smallmatrix}\right)}=
  \psi^k_{\left(\begin{smallmatrix} l-2-k\\l-k \end{smallmatrix}\right)}=
 \begin{pmatrix}    -z &   0 &        xy & y^{\frac{k}{2}} \\
         0 &  -z & y^{l-1-\frac{k}{2}} & -x \\
         x & y^\frac{k}{2} &         z & 0 \\
 y^{l-1-\frac{k}{2}} & -xy &         0 & z \end{pmatrix}\ ,\quad
 k \text{:\ even}\ \ (2\le k\le l-2)
\end{equation*}
\begin{equation*}
  \varphi^k_{\left(\begin{smallmatrix} l-2-k\\l-k \end{smallmatrix}\right)}=
  \psi^k_{\left(\begin{smallmatrix} l-2-k\\l-k \end{smallmatrix}\right)}=
 \begin{pmatrix} -z      & y^{\frac{k-1}{2}} &        xy & 0 \\
   y^{l-\frac{k+1}{2}} &       z &         0 & -x \\
           x &       0 &         z & y^\frac{k-3}{2} \\
           0 &     -xy & y^{l-\frac{k-1}{2}} & -z \end{pmatrix}\ ,\quad
 k \text{:\ odd}\ \ (3\le k\le l-2)
\end{equation*}
}The forms of $(\varphi^{l-1},\psi^{l-1})$ and $(\varphi^l,\psi^l)$
depend on whether $l$ is even or odd.

\noindent
$\bullet$\ If $l$ is even, one obtains
{\small
\begin{equation*}
 \begin{split}
 & \varphi^{l-1}_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}=
  \psi^{l-1}_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}=
 \begin{pmatrix} z                        & y(x+\sqrt{-1} y^{\frac{l-2}{2}}) \\
     x-\sqrt{-1} y^{\frac{l-2}{2}} & -z
 \end{pmatrix}\ ,\\
 & \varphi^l_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}=
  \psi^l_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}=
 \begin{pmatrix} z                        & y(x-\sqrt{-1} y^{\frac{l-2}{2}}) \\
     x+\sqrt{-1} y^{\frac{l-2}{2}} & -z
 \end{pmatrix}\ .
 \end{split}
\end{equation*}
}
\noindent
$\bullet$\ If $l$ is odd, one obtains
{\small
\begin{equation*}
 \begin{split}
  &\varphi^{l-1}_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}
  =\psi^{l}_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}=
\begin{pmatrix} z+\sqrt{-1} y^{\frac{l-1}{2}} & xy \\
                           x & -(z-\sqrt{-1} y^{\frac{l-1}{2}})
 \end{pmatrix}\ ,\\
  &\varphi^l_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}
  =\psi^{l-1}_{\left(\begin{smallmatrix} 1 \end{smallmatrix}\right)}=
\begin{pmatrix} z-\sqrt{-1} y^{\frac{l-1}{2}} & xy \\
                           x & -(z+\sqrt{-1} y^{\frac{l-1}{2}})
 \end{pmatrix}\ .
 \end{split}
\end{equation*}
}

\vspace*{0.3cm}

\noindent
{\bf Type $E_6$.}\ \, The AR-quiver for the polynomial
$f=x^3+y^4+z^2$ is given by:
\begin{equation*}
\xymatrix{
 & & \ [M^1]
 \ar@<0.7ex>[d]^{} & \\
 \ [M^5]\ \ar@<0.5ex>[r]^{}
 & \ [M^3] \ \ar@<0.5ex>[r]^{}
 \ar@<0.5ex>[l]^{}
 & \ [M^2] \ \ar@<0.5ex>[r]^{}
 \ar@<0.5ex>[l]^{} \ar@<0.5ex>[u]^{}
 & \ [M^4] \ \ar@<0.5ex>[r]^{}
 \ar@<0.5ex>[l]^{}
 & \ [M^6] \ \ . \ar@<0.5ex>[l]^{}
}
\end{equation*}
For $Y^\pm:=y^2\pm\sqrt{-1} z$, each pair $(\varphi^k,\psi^k)$ is given by
{\footnotesize
\begin{equation*}
 \varphi^1_{\left(\begin{smallmatrix} 1 \\ 5\end{smallmatrix}\right)}
 =\varphi^1_{\left(\begin{smallmatrix} 1 \\ 5\end{smallmatrix}\right)}
 =\begin{pmatrix}
 -z & 0 & x^2 & y^3 \\
 0 & -z & y & -x \\
 x & y^3 & z & 0  \\
 y & -x^2 & 0 & z
 \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
 \varphi^2_{\left(\begin{smallmatrix} 0\\ 2\\ 4\end{smallmatrix}\right)}=
 \begin{pmatrix} -\sqrt{-1} z &  -y^2 &    xy &     0 &   x^2 & 0 \\
      -y^2 & -\sqrt{-1} z &     0 &     0 &     0 & x \\
         0 &     0 & -\sqrt{-1} z &    -x &     0 & y \\
         0 &    xy &  -x^2 & -\sqrt{-1} z &   y^3 & 0 \\
         x &     0 &     0 &     y & -\sqrt{-1} z & 0 \\
         0 &   x^2 &   y^3 &     0 &  xy^2 & -\sqrt{-1} z
\end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
\psi^2_{\left(\begin{smallmatrix} 0\\ 2\\ 4\end{smallmatrix}\right)}=
 \begin{pmatrix}  \sqrt{-1} z &  -y^2 &    xy &     0 &   x^2 & 0 \\
      -y^2 &  \sqrt{-1} z &     0 &     0 &     0 & x \\
         0 &     0 &  \sqrt{-1} z &    -x &     0 & y \\
         0 &    xy &  -x^2 &  \sqrt{-1} z &   y^3 & 0 \\
         x &     0 &     0 &     y &  \sqrt{-1} z & 0 \\
         0 &   x^2 &   y^3 &     0 &  xy^2 & \sqrt{-1} z
 \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
\varphi^3_{\left(\begin{smallmatrix} 1\\ 3\end{smallmatrix}\right)}=
 \psi^4_{\left(\begin{smallmatrix} 1\\ 3\end{smallmatrix}\right)}=
 \begin{pmatrix} -Y^- &     0 &      xy & x\\
        -xy & Y^+ &     x^2 & 0 \\
          0 &     x & \sqrt{-1} z & y \\
        x^2 &   -xy &     y^3 & \sqrt{-1} z \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
\varphi^4_{\left(\begin{smallmatrix} 1\\ 3\end{smallmatrix}\right)}=
 \psi^3_{\left(\begin{smallmatrix} 1\\ 3\end{smallmatrix}\right)}=
 \begin{pmatrix} -Y^+ &     0 &      xy & x\\
        -xy & Y^- &     x^2 & 0 \\
          0 &     x & -\sqrt{-1} z & y \\
        x^2 &   -xy &     y^3 & -\sqrt{-1} z \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
\varphi^5_{\left(\begin{smallmatrix} 2 \end{smallmatrix}\right)}=
  \psi^6_{\left(\begin{smallmatrix} 2 \end{smallmatrix}\right)}=
  \begin{pmatrix}  -Y^- & x      \\
          x^2 & Y^+ \end{pmatrix}\ ,\qquad
\varphi^6_{\left(\begin{smallmatrix} 2 \end{smallmatrix}\right)}=
  \psi^5_{\left(\begin{smallmatrix} 2 \end{smallmatrix}\right)}=
  \begin{pmatrix}  -Y^+ & x      \\
          x^2 & Y^- \end{pmatrix}\ .
\end{equation*}
}

\vspace*{0.3cm}

\noindent
{\bf Type $E_7$.}\ \, The AR-quiver for the polynomial
$f=x^3+xy^3+z^2$ is given by:
\begin{equation*}
\xymatrix{
 & & & [M^4] \ar@<0.5ex>[d]^{} & & & \\
 \ [M^7] \  \ar@<0.5ex>[r]^{}
 & \ [M^6] \  \ar@<0.5ex>[r]^{}
 \ar@<0.5ex>[l]^{}
 & \ [M^5] \
 \ar@<0.5ex>[r]^{} \ar@<0.5ex>[l]^{}
 & \ [M^3] \  \ar@<0.5ex>[r]^{}
 \ar@<0.5ex>[l]^{} \ar@<0.5ex>[u]^{}
 & \ [M^2] \
\ar@<0.5ex>[r]^{} \ar@<0.5ex>[l]^{}
 & \ [M^1] \ .
\ar@<0.5ex>[l]^{}
 &
}
\end{equation*}
The corresponding matrix factorizations are:
{\footnotesize
\begin{equation*}
 \varphi^1_{\left(\begin{smallmatrix} 2\\8\end{smallmatrix} \right)}
 = \psi^1_{\left(\begin{smallmatrix} 2\\8\end{smallmatrix} \right)}
 =\begin{pmatrix} z & 0 & -x^2 & y \\
      0 & z & xy^2 & x \\
      -x & y & -z & 0 \\
      xy^2 & x^2 & 0 & -z \end{pmatrix}\ ,\quad
 \varphi^4_{\left(\begin{smallmatrix} 1\\5\end{smallmatrix} \right)}
 = \psi^4_{\left(\begin{smallmatrix} 1\\5\end{smallmatrix} \right)}
 =\begin{pmatrix} -z & y^2 & 0 & x \\
      xy & z & -x^2 & 0\\
      0 & -x & -z & y \\
      x^2 & 0 & xy^2 & z \end{pmatrix}\
\end{equation*}
\begin{equation*}
 \varphi^2_{\left(\begin{smallmatrix} 1\\3\\7\end{smallmatrix} \right)}
 = \psi^2_{\left(\begin{smallmatrix} 1\\3\\7\end{smallmatrix} \right)}
 =\begin{pmatrix} -z & y^2 & xy & 0 & x^2 & 0 \\
      xy & z & 0 & 0 & 0 & -x\\
      0 & 0 & z & -x & 0 & y \\
      0 & -xy & -x^2 & -z & xy^2 & 0 \\
      x & 0 & 0 & y & z & 0 \\
      0 & -x^2 & xy^2 & 0 & x^2y & -z \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
 \varphi^5_{\left(\begin{smallmatrix} 1\\3\\5\end{smallmatrix} \right)}
 =\psi^5_{\left(\begin{smallmatrix} 1\\3\\5\end{smallmatrix} \right)}
 =\begin{pmatrix} -z & 0 & xy & 0 & 0 & x \\
      -xy & z & 0 & -y^2 & -x^2 & 0\\
      y^2 & 0 & z & -x & xy & 0 \\
      0 & -xy & -x^2 & -z & 0 & 0 \\
      0 & -x & 0 & 0 & -z & -y \\
      x^2 & 0 & 0 & xy & -xy^2 & z \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
 \varphi^3_{\left(\begin{smallmatrix} 0\\2\\4\\6\end{smallmatrix} \right)}
 = \psi^3_{\left(\begin{smallmatrix} 0\\2\\4\\6\end{smallmatrix} \right)}
 =\begin{pmatrix}
 -z & 0 & xy & -y^2 & 0 & 0 & x^2 & 0 \\
 0 & -z & 0 & y^2 & 0 & 0 & 0 & x \\
 y^2 & y^2 & z & 0 & 0 & -x & 0 & 0\\
 0 & xy & 0 & z & -x^2 & 0 & 0 & 0\\
 0 & 0 & 0 & -x & -z & 0 & 0 & y \\
 0 & 0 & -x^2 & 0 & 0 & -z & xy^2 & y^2 \\
 x & 0 & 0 & 0 & -y^2 & y^2 & z & 0\\
 0 & x^2 & 0 & 0 & xy^2 & 0 & 0 & z
\end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
 \varphi^6_{\left(\begin{smallmatrix} 2\\4\end{smallmatrix} \right)}
 = \psi^6_{\left(\begin{smallmatrix} 2\\4\end{smallmatrix} \right)}
 =\begin{pmatrix} z & 0 & -xy & x \\
      0 & z & x^2 & y^2\\
      -y^2 & x & -z & 0 \\
      x^2 & xy & 0 & -z \end{pmatrix}\ ,\quad
 \varphi^7_{\left(\begin{smallmatrix} 3\end{smallmatrix} \right)}
 = \psi^7_{\left(\begin{smallmatrix} 3\end{smallmatrix} \right)}
 =\begin{pmatrix} z & x \\
      x^2+y^3 & -z \end{pmatrix}\ .
\end{equation*}
}

\vspace*{0.3cm}

\noindent
{\bf Type $E_8$.}\ \,
The AR-quiver for the polynomial $f=x^3+y^5+z^2$ is given by:
\begin{equation*}
\hspace*{-0.7cm}
\xymatrix{
 & & & & [M^6] \ar@<0.5ex>[d]^{} & & \\
 \ [M^1] \  \ar@<0.5ex>[r]^{}
 & \ [M^2] \
 \ar@<0.5ex>[r]^{} \ar@<0.5ex>[l]^{}
 & \ [M^3] \
 \ar@<0.5ex>[r]^{} \ar@<0.5ex>[l]^{}
 & \ [M^4] \
\ar@<0.5ex>[r]^{} \ar@<0.5ex>[l]^{}
 & \ [M^5] \
\ar@<0.5ex>[r]^{} \ar@<0.5ex>[l]^{} \ar@<0.5ex>[u]^{}
 & \ [M^7] \
\ar@<0.5ex>[r]^{} \ar@<0.5ex>[l]^{}
 & \ [M^8] \ .
\ar@<0.5ex>[l]^{}
}
\end{equation*}
The corresponding matrix factorizations are :
{\scriptsize
\begin{equation*}
 \hspace*{-0.5cm}
 \varphi^1_{\left(\begin{smallmatrix} 4\\14\end{smallmatrix} \right)}
 = \psi^1_{\left(\begin{smallmatrix} 4\\14\end{smallmatrix} \right)}
 =\begin{pmatrix} z & 0 & x & y \\
      0 & z & y^4 & -x^2\\
      x^2 & y & -z & 0 \\
      y^4 & -x & 0 & -z \end{pmatrix}\ ,\quad
 \varphi^8_{\left(\begin{smallmatrix} 2\\8\end{smallmatrix} \right)}
 = \psi^8_{\left(\begin{smallmatrix} 2\\8\end{smallmatrix} \right)}
 =\begin{pmatrix} z & 0 & x & y^2 \\
      0 & z & y^3 & -x^2\\
      x^2 & y^2 & -z & 0 \\
      y^3 & -x & 0 & -z \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
 \varphi^2_{\left(\begin{smallmatrix} 3\\5\\13\end{smallmatrix} \right)}
 = \psi^2_{\left(\begin{smallmatrix} 3\\5\\13\end{smallmatrix} \right)}
 =\begin{pmatrix}
 z & -y^2 & xy & 0 & -x^2 & 0 \\
 -y^3 & -z & 0 & 0 & 0 & x \\
 0 & 0 & -z & x & 0 & y \\
 0 & -xy & x^2 & z & y^4 & 0\\
 -x & 0 & 0 & y & -z & 0 \\
 0 & x^2 & y^4 & 0 & -xy^3 & z
 \end{pmatrix}\ ,
 \ \
 \varphi^6_{\left(\begin{smallmatrix} 1\\5\\9\end{smallmatrix} \right)}
 =\psi^6_{\left(\begin{smallmatrix} 1\\5\\9\end{smallmatrix} \right)}
 =\begin{pmatrix}
 -z & 0 & 0 & y^2 & 0 & x \\
 xy & z & -y^3 & 0 & -x^2 & 0 \\
 0 & -y^2 & -z & x & 0 & 0 \\
 y^3 & 0 & x^2 & z & -xy^2 & 0\\
 0 & -x & 0 & 0 & -z & y \\
 x^2 & 0 & -xy^2 & 0 & y^4 & z
 \end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
 \varphi^3_{\left(\begin{smallmatrix} 2\\4\\6\\12\end{smallmatrix} \right)}
 = \psi^3_{\left(\begin{smallmatrix} 2\\4\\6\\12\end{smallmatrix} \right)}
 =\begin{pmatrix}
 -z & 0 & -xy & y^2 & 0 & 0 & x^2 & 0 \\
 0 & -z & y^3 & 0 & 0 & 0 & 0 & x \\
 0 & y^2 & z & 0 & 0 & -x & 0 & 0\\
 y^3 & xy & 0 & z & -x^2 & 0 & 0 & 0\\
 0 & 0 & 0 & -x & -z & 0 & y^3 & y \\
 0 & 0 & -x^2 & 0 & 0 & -z & 0 & y^2 \\
 x & 0 & 0 & 0 & y^2 & -y & z & 0\\
 0 & x^2 & 0 & 0 & 0 & y^3 & 0 & z
\end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
 \varphi^4_{\left(\begin{smallmatrix} 1\\3\\5\\7\\11\end{smallmatrix} \right)}
= \psi^4_{\left(\begin{smallmatrix} 1\\3\\5\\7\\11\end{smallmatrix} \right)}
 =\begin{pmatrix}
 z & 0 & xy & 0 & 0 & -y^2 & y^3 & 0 & -x^2 & 0 \\
 0 & -z & 0 & 0 & 0 & 0 & 0 & -y^2 & 0 & x \\
 0 & 0 & -z & y^2 & 0 & 0 & 0 & x & 0 & 0\\
 0 & xy & y^3 & z & 0 & 0 & -x^2 & 0 & 0 & 0\\
 0 & y^2 & 0 & 0 & z & -x & 0 & 0 & y^3 & 0 \\
 -y^3 & 0 & 0 & 0 & -x^2 & -z & 0 & 0 & 0 & y^2 \\
 0 & 0 & 0 & -x & 0 & 0 & -z & 0 & 0 & y\\
 0 & -y^3 & x^2 & 0 & 0 & 0 & xy^2 & z & 0 & 0 \\
 -x & 0 & 0 & 0 & y^2 & 0 & 0 & y & -z & 0\\
 0 & x^2 & xy^2 & 0 & 0 & 0 & y^4 & 0 & 0 & z
\end{pmatrix}\ ,
\end{equation*}
\begin{equation*}
\hspace*{-0.8cm}
 \varphi^5_{\left(\begin{smallmatrix} 0\\2\\4\\6\\8\\10\end{smallmatrix} \right)}
 = \psi^5_{\left(\begin{smallmatrix} 0\\2\\4\\6\\8\\10\end{smallmatrix} \right)}
 =
 \left(
 \begin{array}{cccccccccccc}
 -z & 0 & 0 & 0 & 0 & 0 & 0 & y^2 & 0 & 0 & 0 & x \\
 0 & -z & -xy & 0 & 0 & 0 & y^3 & -y^2 & 0 & 0 & x^2 & 0 \\
 0 & 0 & z & 0 & 0 & -y^2 & 0 & 0 & y^3 & -x & 0 & 0 \\
 xy & 0 & 0 & z & -y^3 & 0 & 0 & 0 & -x^2 & 0 & 0 & 0 \\
 0 & 0 & 0 & -y^2 & -z & 0 & 0 & x & 0 & 0 & 0 & 0 \\
 0 & 0 & -y^3 & 0 & 0 & -z & -x^2 & 0 & 0 & 0 & xy^2 & y^2 \\
 y^2 & y^2 & 0 & 0 & 0 & -x & z & 0 & 0 & 0 & 0 & 0 \\
 y^3 & 0 & 0 & 0 & x^2 & 0 & 0 & z & -xy^2 & 0 & 0 & 0 \\
 0 & 0 & 0 & -x & 0 & 0 & 0 & 0 & -z & 0 & 0 & y \\
 0 & 0 & -x^2 & -y^3 & 0 & 0 & xy^2 & 0 & 0 & -z & -y^4 & 0 \\
 0 & x & 0 & 0 & y^2 & 0 & 0 & 0 & 0 & -y & z & 0 \\
 x^2 & 0 & 0 & 0 & -xy^2 & 0 & 0 & 0 & y^4 & 0 & 0 & z
 \end{array}\right).
\end{equation*}
\begin{equation*}
 \varphi^7_{\left(\begin{smallmatrix} 1\\3\\7\\9\end{smallmatrix} \right)}
 =\psi^7_{\left(\begin{smallmatrix} 1\\3\\7\\9\end{smallmatrix} \right)}
 =\begin{pmatrix}
 z & 0 & 0 & 0 & -y^3 & 0 & 0 & -x \\
 xy & -z & 0 & 0 & 0 & y^2 & x^2 & 0 \\
 0 & 0 & -z & y^2 & 0 & x & -y^3 & 0\\
 0 & 0 & 0 & z & -x^2 & 0 & 0 & y^2\\
 -y^2 & 0 & 0 & -x & -z & 0 & 0 & 0 \\
 0 & y^3 & x^2 & 0 & xy^2 & z & 0 & 0 \\
 0 & x & -y^2 & 0 & 0 & 0 & z & y\\
 -x^2 & 0 & 0 & y^3 & 0 & 0 & 0 & -z
\end{pmatrix}\ .
\end{equation*}
}
The shift functor $T$ acts on these matrix factorizations as follows.
\begin{itemize}
 \item
For type $A_l$, $T(M^k)\simeq M^{l+2-k}$ for any $k\in\Pi$.
 \item
For type $D_l$, $T(M^k)\simeq M^k$ for all $k\in\Pi$,
except that $T(M^{l-1})\simeq M^l$, $T(M^l)\simeq M^{l-1}$
if $l$ is odd.
 \item
For type $E_6$, $T(M^k)\simeq M^k$ for $k=1,2$ but $T(M^3)\simeq M^4$,
$T(M^4)\simeq M^3$ and
$T(M^5)\simeq M^6$, $T(M^6)\simeq M^5$.
 \item
For type $E_7$ or $E_8$, $T(M^k)\simeq M^k$ for any $k\in\Pi$.
\end{itemize}
Remember that the Serre functor
(in Theorem \ref{thm:grading} (iii))
is defined by ${\mathcal S}:=T\tau^{-1}$ when we shall introduce the grading.
 \label{tbl:1}
\end{tbl}

\begin{tbl}
The list of all the isomorphism classes of the indecomposable objects
in $\tHMF{R}$ for a polynomial $f\in R$ of type ADE is given.

The set of the isomorphism classes of the indecomposable objects
is given by
\begin{equation*}
\left\{[\ti{M}^k_n:=(Q^k,S^k_n)],
\quad k\in\Pi,\quad n\in{\mathbb Z} \right\}\ .
\end{equation*}
Here, for each $k\in\Pi$, $Q^k$ is the matrix factorization
of size $2\nu_k$ given in Table \ref{tbl:1},
and the grading matrix $S^k_n$ is a diagonal matrix as follows:
\begin{equation*}
 S^k_n:
=\mathrm{diag}\left(q^k_1,-q^k_1,\cdots,q^k_{\nu_k},-q^k_{\nu_k};
 {\bar q}^k_1,-{\bar q}^k_1,\cdots,{\bar q}^k_{\nu_k},-{\bar q}^k_{\nu_k}\right)
+\phi^k_n\cdot{\bf 1}_{4\nu_k}\ ,
\end{equation*}
where the phase is given by
$\phi^k_n=\frac{2n+\sigma}{h}$ for $k\in\Pi_\sigma$, $\sigma=1,2$,
and the data $q^k_j,{\bar q}^k_j$ are given below.

 \vspace*{0.3cm}

\noindent
{\bf Type $A_l$}\ $(h=l+1)$:
\quad
In this case, $\nu_k=1$ for all $k\in\Pi$
and the grading is given by
\begin{equation*}
(q^k_1;{\bar q}^k_1)
 =\ov{l+1}(b-k;(l+1-b)-k)\ .
\end{equation*}

 \vspace*{0.3cm}

For type $D_l$ and $E_l$, for any $k\in\Pi$
there exists a representative of
the isomorphism classes of the indecomposable objects such that
$q^k_j={\bar q}^k_j$, $j=1,\cdots,\nu_k$.
The matrix factorizations $Q^k$'s listed in Table \ref{tbl:1}
are just such ones.
Therefore, we present only $q^k_j$ and omit ${\bar q}^k_j$.

 \vspace*{0.3cm}

\noindent
{\bf Type $D_l$}\ $(h=2(l-1))$:
\begin{minipage}[t]{80mm}{
\begin{equation*}
 \begin{array}{lll}
 k & \nu_k & (q^k_1,\cdots,q^k_{\nu_k}) \\
 1 &   1 & \ov{2(l-1)}(l-3) \\
 2,\cdots,(l-2)& 2 & \ov{2(l-1)}(l-k-2,l-k) \\
 (l-1),l & 1 & \ov{2(l-1)}(1) \ .\\
 \end{array}
\end{equation*}
}
\end{minipage}
 \vspace*{0.3cm}

\noindent

\begin{equation*}
 \begin{array}{lll}
 \text{{\bf Type $E_6$}\ $(h=12)$: }\qquad\quad
 & \text{{\bf Type $E_7$}\ $(h=18)$: }\qquad\quad
 & \text{{\bf Type $E_8$}\ $(h=30)$: }\qquad \\
 \begin{array}{llc}
 k & \nu_k & (q^k_1,\cdots,q^k_{\nu_k}) \\
 1 & 2 & \ov{12}(1,5)\ \\
 2 & 3 & \ov{12}(0,2,4)\ \\
 3,4 & 2 & \ov{12}(1,3)\ \\
 5,6 & 1 & \ov{12}(2)\ .
 \end{array}
 &
 \begin{array}{llc}
 k & \nu_k & (q^k_1,\cdots,q^k_{\nu_k}) \\
 1 & 2   & \ov{18}(2,8)\ \\
 2 & 3   & \ov{18}(1,3,7)\ \\
 3 & 4   & \ov{18}(0,2,4,6)\ \\
 4 & 2   & \ov{18}(1,5)\ \\
 5 & 3   & \ov{18}(1,3,5)\ \\
 6 & 2   & \ov{18}(2,4)\ \\
 7 & 1   & \ov{18}(3)\ .
 \end{array}
 &
 \begin{array}{ccc}
 k & \nu_k & (q^k_1,\cdots,q^k_{\nu_k}) \\
 1 & 2 & \ov{30}(4,14)\ \\
 2 & 3 & \ov{30}(3,5,13)\ \\
 3 & 4 & \ov{30}(2,4,6,12)\ \\
 4 & 5 & \ov{30}(1,3,5,7,11)\ \\
 5 & 6 & \ov{30}(0,2,4,6,8,10)\ \\
 6 & 3 & \ov{30}(1,5,9)\ \\
 7 & 4 & \ov{30}(1,3,7,9)\ \\
 8 & 2 & \ov{30}(2,8)\ .
 \end{array}
 \end{array}
\end{equation*}
\begin{enumerate}
 \item If we set $n=0$ for all $k\in\Pi$, then
$\{\ti{M}^1_0,\cdots,\ti{M}^l_0\}$ corresponds to the Dynkin quiver
with a principal orientation (see Proposition \ref{prop:abelian}).

 \item For the grading matrix of $\ti{M}^k_n$,
 one has $q^k_j\ne 0$ for $j>1$, and $q^k_1=0$ if
 and only if $F(\ti{M}^k_n)=M_o$.

 \item Lemma \ref{lem:B} can be checked
at the level of the grading matrices $S$. Namely,
for an indecomposable object $\ti{M}^k_n\in\tHMF{R}$,
the cone $C(\Psi)$ of a nonzero morphism
$\Psi\in\HomDz{R}({\mathcal S}^{-1}(\ti{M}^k_n),\ti{M}^k_n)$
is isomorphic to
the direct sum of indecomposable objects
$\ti{M}^{k_1}_{n_1}\oplus\cdots\oplus\ti{M}^{k_m}_{n_m}$
for some $m\in{\mathbb Z}_{>0}$
such that $\{k_1,\cdots,k_m\}=\{k'\in\Pi\ |\ d(k,k')=1\}$ and
$\phi(\ti{M}^{k_i}_{n_i})=\phi(\ti{M}^k_n)+\ov{h}$
for any $i=1,\cdots,m$.
Correspondingly, one can check that
\begin{align}
 & \left\{q^k_1-\ov{h},-q^k_1-\ov{h},\cdots,q^k_{\nu_k}-\ov{h},
 -q^k_{\nu_k}-\ov{h}\right\}
\coprod
\left\{q^k_1+\ov{h},-q^k_1+\ov{h},\cdots,q^k_{\nu_k}+\ov{h},
 -q^k_{\nu_k}+\ov{h}\right\} \nonumber\\
 &
 \qquad=\coprod_{i=1}^m
 \left\{q^{k_i}_1,-q^{k_i}_1,\cdots,
 q^{k_i}_{\nu_{k_i}},-q^{k_i}_{\nu_{k_i}}\right\}  \label{cone-grading}
\end{align}
\end{enumerate}
 \label{tbl:2}
\end{tbl}
and the same identities for ${\bar q}^k_j$, $k\in\Pi$, $j=1,\cdots,\nu_k$.

\begin{tbl}\label{tbl:3}
The following tables give all the morphisms between
all the isomorphism classes of the indecomposable objects in $\tHMF{R}$
for a polynomial $f$ of type ADE.

Recall that,
for two indecomposable objects $\ti{M}^k_n,\ti{M}^{k'}_{n'}\in\tHMF{R}$,
$\frak C(k,k')$ is the multi-set of non-negative integers such that
\begin{equation*}
\frak C(k,k')
 :=\{c:=h(\phi^{k'}_{n'}-\phi^k_n)+2n''\ |\
 \HomDz{R}(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'}))\ne 0\ ,\ \ n\in{\mathbb Z}\}\ ,
\end{equation*}
where the integer $c$ appears with multiplicity
$d:=\dim_{\mathbb C}(\HomDz{R}(\ti{M}^k_n,\tau^{n''}(\ti{M}^{k'}_{n'})))$.
We sometimes write $c^d$ instead of $d$ copies of $c$.

For each type of ADE, $\frak C(k,k')$ is listed up for any
$k,k'\in\Pi$.

\vspace*{0.3cm}

\noindent
{\bf Type $A_l$}\ $(h-2=l-1)$\ : \
One obtains (essentially the same as those given in
\cite{hw:1})
\begin{equation*}
 \frak C(k,k')=\left\{
 \begin{split}
 & |k'-k|,\ |k'-k|+2,\ |k'-k|+4,\ \cdots\cdots\ , \\
 & \hspace*{1.5cm}\cdots\cdots\ ,l-3-|(l-1)-(k+k'-2)|,\ \,
 l-1-|(l-1)-(k+k'-2)|
 \end{split}
 \right\}\ .
\end{equation*}
For any $k,k'\in\Pi$,
$\frak C(k,k')$ does not contain multiple copies of the same integer.
Pictorially, the table of $\frak C(k,k')$ is displayed as
\begin{equation*}
 \renewcommand{\arraystretch}{1.5}
\begin{array}{@{\vrule width 1pt\ \ }c|c|c|c|c|c|c|c@{\ \vrule width 1pt}}
  \hline
 \overset{}{k}\backslash \underset{}{k'} &
 1 & 2 & 3 & \cdots\cdots\cdots & l-2 & l-1 & l \\
  \hline
 1 &
 \begin{smallmatrix} 0 \end{smallmatrix} &
 \begin{smallmatrix} 1 \end{smallmatrix} &
 \begin{smallmatrix} 2 \end{smallmatrix} &
 \begin{smallmatrix} \cdots\cdots\cdots \end{smallmatrix} &
 \begin{smallmatrix} l-3 \end{smallmatrix} &
 \begin{smallmatrix} l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-1 \end{smallmatrix} \\
  \hline
 2 &
 \begin{smallmatrix} 1 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 \end{smallmatrix} &
 \begin{smallmatrix} \cdots\cdots\cdots \end{smallmatrix} &
 \begin{smallmatrix} l-4 & l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-3 & l-1 \end{smallmatrix} &
 \begin{smallmatrix} l-2 \end{smallmatrix} \\
  \hline
 3 &
 \begin{smallmatrix} 2 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 & 4 \end{smallmatrix} &
 \begin{smallmatrix} \cdots\cdots\cdots \end{smallmatrix} &
 \begin{smallmatrix} l-5 & l-3 & l-1 \end{smallmatrix} &
 \begin{smallmatrix} l-4 & l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-3 \end{smallmatrix} \\
 \hline
 \vdots &
 \begin{smallmatrix} \vdots\\ \vdots\\ \vdots \end{smallmatrix} &
 \begin{smallmatrix} \vdots\\ \vdots\\ \vdots \end{smallmatrix} &
 \begin{smallmatrix} \vdots\\ \vdots\\ \vdots \end{smallmatrix} &
 \begin{smallmatrix}  && \\ & \ddots & \\ && \end{smallmatrix} &
 \begin{smallmatrix} \vdots\\ \vdots\\ \vdots \end{smallmatrix} &
 \begin{smallmatrix} \vdots\\ \vdots\\ \vdots \end{smallmatrix} &
 \begin{smallmatrix} \vdots\\ \vdots\\ \vdots \end{smallmatrix} \\
  \hline
 l-2 &
 \begin{smallmatrix} l-3 \end{smallmatrix} &
 \begin{smallmatrix} l-4 & l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-5 & l-3 & l-1 \end{smallmatrix} &
 \begin{smallmatrix} \cdots\cdots\cdots \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 & 4 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 \end{smallmatrix} &
 \begin{smallmatrix} 2 \end{smallmatrix} \\
  \hline
 l-1 &
 \begin{smallmatrix} l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-3 & l-1 \end{smallmatrix} &
 \begin{smallmatrix} l-4 & l-2 \end{smallmatrix} &
 \begin{smallmatrix} \cdots\cdots\cdots \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 \end{smallmatrix} &
 \begin{smallmatrix} 1 \end{smallmatrix} \\
  \hline
 l &
 \begin{smallmatrix} l-1 \end{smallmatrix} &
 \begin{smallmatrix} l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-3 \end{smallmatrix} &
 \begin{smallmatrix} \cdots\cdots\cdots \end{smallmatrix} &
 \begin{smallmatrix} 2 \end{smallmatrix} &
 \begin{smallmatrix} 1 \end{smallmatrix} &
 \begin{smallmatrix} 0 \end{smallmatrix} \\
  \hline
 \end{array}
\end{equation*}

\vspace*{0.3cm}

\noindent
{\bf Type $D_l$}\ $(h-2=2l-4)$\ :
\begin{equation*}
 \renewcommand{\arraystretch}{2.5}
\begin{array}{@{\vrule width 1pt\ \ }c|c|c|c|c@{\ \vrule width 1pt}}
  \hline
 k\backslash k' & 1 & k'\ (\begin{smallmatrix} 2\le k'\le l-2\end{smallmatrix} ) & l-1 & l \\
  \hline
 1 &
 \begin{smallmatrix} 0 & 2l-4 \end{smallmatrix} &
 \begin{smallmatrix} k'-1 & 2l-3-k' \end{smallmatrix} &
 \begin{smallmatrix} l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-2 \end{smallmatrix} \\
  \hline
\underset{(2\le k\le l-2)}{k} &
 \begin{smallmatrix} k-1\\ 2l-3-k \end{smallmatrix} &
 \begin{smallmatrix} |k'-k| \ \, |k'-k|+2 \ \, \cdots\cdots \quad  \\
 \quad \cdots\cdots \ \, k+k'-4 \ \, k+k'-2\ , \\
 2l-2-(k+k') \ 2l-(k+k') \ \cdots\cdots\quad \\
 \quad\cdots\cdots \ \, 2l-6-|k'-k| \ \, 2l-4-|k'-k| \\ \end{smallmatrix} &
 \begin{smallmatrix} l-1-k & l+1+k & \cdots \\
      \cdots & l-5+k & l-3+k \end{smallmatrix} &
 \begin{smallmatrix} l-1-k & l+1+k & \cdots \\
      \cdots & l-5+k & l-3+k \end{smallmatrix} \\
  \hline
 l-1 &
 \begin{smallmatrix} l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-1-k & l+1+k & \cdots \\
      \cdots & l-5+k & l-3+k \end{smallmatrix} &
 \begin{smallmatrix} 0 & 4 & 8 & \cdots & 2l-4 \\
        &   &   &       (l:& \text{even}) \\
      0 & 4 & 8 & \cdots & 2l-6 \\
        &   &   &       (l:& \text{odd})
 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 6 & 10 & \cdots & 2l-6 \\
         &   &   &       (l:& \text{even}) \\
      2 & 6 & 10 & \cdots & 2l-4 \\
        &   &   &       (l:& \text{odd})
      \end{smallmatrix} \\
  \hline
 l &
 \begin{smallmatrix} l-2 \end{smallmatrix} &
 \begin{smallmatrix} l-1-k & l+1+k & \cdots \\
      \cdots & l-5+k & l-3+k \end{smallmatrix} &
 \begin{smallmatrix} 2 & 6 & 10 & \cdots & 2l-6 \\
         &   &   &       (l:& \text{even}) \\
      2 & 6 & 10 & \cdots & 2l-4 \\
        &   &   &       (l:& \text{odd})
 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 4 & 8 & \cdots & 2l-4 \\
        &   &   &       (l:& \text{even}) \\
      0 & 4 & 8 & \cdots & 2l-6 \\
        &   &   &       (l:& \text{odd})
 \end{smallmatrix} \\
  \hline
 \end{array}
\end{equation*}

\vspace*{0.3cm}

\noindent
{\bf Type $E_6$}\ $(h-2=10)$\ :
\begin{equation*}
 \renewcommand{\arraystretch}{1.5}
\begin{array}{@{\vrule width 1pt\ \ }c|c|c|c|c|c|c@{\ \vrule width 1pt}}
  \hline
 k\backslash k' & 1 & 2 & 3 & 4 & 5 & 6 \\
  \hline
 1 &
 \begin{smallmatrix} 0 & 4 & 6 & 10 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 & 5^2 & 7 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 3 & 7 \end{smallmatrix} &
 \begin{smallmatrix} 3 & 7 \end{smallmatrix} \\
  \hline
 2 &
  \begin{smallmatrix} 1 & 3 & 5^2 & 7 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2^2 & 4^3 & 6^3 & 8^2 & 10 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^2 & 7^2 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^2 & 7^2 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} \\
  \hline
 3 &
  \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^2 & 7^2 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 & 4 & 6^2 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4^2 & 6 & 8 & 10 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 5 & 7 \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 9 \end{smallmatrix} \\
  \hline
 4 &
 \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^2 & 7^2 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4^2 & 6 & 8 & 10 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 & 4 & 6^2 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 5 & 7 \end{smallmatrix} \\
  \hline
 5 &
 \begin{smallmatrix} 3 & 7 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 5 & 7 \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 6 \end{smallmatrix} &
 \begin{smallmatrix} 4 & 10 \end{smallmatrix} \\
  \hline
 6 &
 \begin{smallmatrix} 3 & 7 \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6 & 8 \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 9 \end{smallmatrix} &
 \begin{smallmatrix} 1 & 5 & 7 \end{smallmatrix} &
 \begin{smallmatrix} 4 & 10 \end{smallmatrix} &
 \begin{smallmatrix} 0 & 6 \end{smallmatrix}  \\
  \hline
 \end{array}
\end{equation*}

\vspace*{0.3cm}

\noindent
{\bf Type $E_7$}\ $(h-2=16)$\ :
\begin{equation*}
 \renewcommand{\arraystretch}{2.0}
\begin{array}{@{\vrule width 1pt\ \ }c|c|c|c|c|c|c|c@{\ \vrule width 1pt}}
  \hline
 k\backslash k' & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
  \hline
 1 &
 \begin{smallmatrix} 0 & 6 & 10 & 16 \\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 1 & 5 & 7 \\ 9 & 11 & 15\\ && \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6 \\ &8^2& \\ 10 & 12 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 3 & 7 & 9 & 13\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 7 \\ 9 & 11 & 13\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 4 & 6 & 10 & 12\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 5 & 11\\ & \end{smallmatrix} \\
  \hline
 2 &
 \begin{smallmatrix} 1 & 5 & 7 \\ 9 & 11 & 15\\ && \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 & 4\\ 6^2 & 8^2 & 10^2\\ 12 & 14 & 16\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^2 & 7^3 \\ 9^3 & 11^2 & 13^2 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6\\ &8^2& \\ 10 & 12 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4^2 & 6^2\\ &8^2& \\ 10^2 & 12^2 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5^2 & 7 \\ 9 & 11^2 & 13\\ && \end{smallmatrix} &
 \begin{smallmatrix} 4 & 6 & 10 & 12\\ &&& \end{smallmatrix} \\
  \hline
 3 &
 \begin{smallmatrix} 2 & 4 & 6\\ &8^2& \\ 10 & 12 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^2 & 7^3 \\ 9^3 & 11^2 & 13^2 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2^2 & 4^3 \\ 6^4 & 8^4 & 10^4 \\ 12^3 & 14^2 & 16\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 & 5^2 & 7^2 \\ 9^2 & 11^2 & 13 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^3 & 7^3 \\ 9^3 & 11^3 & 13^2 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4^2 & 6^2\\ & 8^2& \\ 10^2 & 12^2 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 7 \\ 9 & 11 & 13\\ && \end{smallmatrix} \\
  \hline
 4 &
 \begin{smallmatrix} 3 & 7 & 9 & 13\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6\\ &8^2& \\ 10 & 12 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 & 5^2 & 7^2 \\ 9^2 & 11^2 & 13 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 0 & 4 & 6 \\ &8& \\ 10 & 12 & 16\\ && \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6^2\\ &8& \\ 10^2 & 12 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 7 \\ 9 & 11 & 13\\ && \end{smallmatrix} &
 \begin{smallmatrix} 4 & 8 & 12\\ && \end{smallmatrix} \\
  \hline
 5 &
 \begin{smallmatrix} 3 & 5 & 7 \\ 9 & 11 & 13\\ && \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4^2 & 6^2\\ &8^2& \\ 10^2 & 12^2 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3^2 & 5^3 & 7^3 \\ 9^3 & 11^3 & 13^2 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4 & 6^2\\ &8& \\ 10^2 & 12 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 & 4^2 \\ 6^2 & 8^3 & 10^2 \\ 12^2 & 14 & 16\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 & 5 & 7^2 \\ 9^2 & 11 & 13 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} \, 2\ \ 6\\ 8 \\ 10\ 14\\ \ \end{smallmatrix} \\
  \hline
 6 &
 \begin{smallmatrix} 4 & 6 & 10 & 12\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5^2 & 7 \\ 9 & 11^2 & 13\\ && \end{smallmatrix} &
 \begin{smallmatrix} 2 & 4^2 & 6^2\\ & 8^2& \\ 10^2 & 12^2 & 14\\ && \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 7 \\ 9 & 11 & 13\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 3 & 5 & 7^2 \\ 9^2 & 11 & 13 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 0 & 2 & 6\\ &8^2& \\ 10 & 14 & 16\\ && \end{smallmatrix} &
 \begin{smallmatrix} 1 & 7 & 9 & 15\\ &&& \end{smallmatrix} \\
  \hline
 7 &
 \begin{smallmatrix} 5 & 11\\ & \end{smallmatrix} &
 \begin{smallmatrix} 4 & 6 & 10 & 12\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 3 & 5 & 7 \\ 9 & 11 & 13\\ && \end{smallmatrix} &
 \begin{smallmatrix} 4 & 8 & 12\\ && \end{smallmatrix} &
 \begin{smallmatrix} \, 2\ \ 6\\ 8 \\ 10\ 14\\ \ \end{smallmatrix} &
 \begin{smallmatrix} 1 & 7 & 9 & 15\\ &&& \end{smallmatrix} &
 \begin{smallmatrix} 0 & 8 & 16\\ && \end{smallmatrix} \\
  \hline
\end{array}
\end{equation*}

\vspace*{0.3cm}

\noindent
{\bf Type $E_8$}\ $(h-2=28)$\ :
\begin{equation*}
 \renewcommand{\arraystretch}{2.5}
\begin{array}{@{\vrule width 1pt\ \ }c|c|c|c|c|c|c|c|c@{\ \vrule width 0.1pt}}
  \hline
 k\backslash k' & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
  \hline
 1 &
 \begin{smallmatrix} 0 & 10 \\ 18 & 28 \\ & \end{smallmatrix} &
 \begin{smallmatrix} 1 & 9 & 11 \\ 17 & 19 & 27 \\ & & \end{smallmatrix} &
 \begin{smallmatrix} \ 2\ \ 8 \ \, 10 \\ 12\ 16 \\ 18\ 20\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 3 \ \ 7 \ \ 9 \\ 11\ 13 \\ 15\ 17 \\ 19\ 21\ 25  \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 4\ \ 6\\ \, 8\ \, 10\\
 12\ 14^2\, 16\\ 18\ 20\\ 22\ 24  \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 5 & 9 & 13 \\ 15 & 19 & 23 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 5\ \ 7 \ \ 11 \\ 13\ 15 \\ 17\ 21\ 23  \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 6 & 12 \\ 16 & 22 \\ \\ \end{smallmatrix} \\
  \hline
 2 &
 \begin{smallmatrix} 1 & 9 & 11 \\ 17 & 19 & 27 \\ \end{smallmatrix} &
\begin{smallmatrix} 0\ \ 2\ \ 8 \\ 10^2\ \, 12\\
 \, 16\ \ \, 18^2 \\ 20\ 26\ 28 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 1\! & 3\! & 7 \\ 9^2\! & 11^2\! & 13 \\
 15\! & 17^2\! & 19^2 \\ 21\! & 25\! & 27 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix}  2\ \ 4\ \ 6 \\ 8^2\ \, 10^2 \\ 12^2\, 14^2\, 16^2 \\
       18^2\ 20^2 \\ 22\ 24\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5^2\! & 7^2\\ 9^2\! & 11^2\! & 13^3 \\
      15^3\! & 17^2\! & 19^2\! \\ 21^2\! & 23^2\! & 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 4\ \ 6 \\ \, 8\ \, 10 \\ 12\ 14^2\, 16 \\
 18\ 20 \\ 22\ 24 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \ 4\ \ 6^2 \\ \, 8\ \, 10 \\ 12^2\, 14^2\, 16^2 \\
 18\ 20 \\ 22^2\ 24 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 5\ \ 7\ \ 11 \\ 13\ 15 \\ 17\ 21\ 23 \\ \\ \end{smallmatrix} \\
  \hline
 3 &
 \begin{smallmatrix} \ 2\ \ 8\ \, 10 \\ 12\ 16 \\ 18\ 20\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 1\! & 3\! & 7 \\ 9^2\! & 11^2\! & 13 \\
 15\! & 17^2\! & 19^2 \\ 21\! & 25\! & 27 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 0\! & 2\! & 4 \\ 6\! & 8^2\! & 10^3 \\ 12^2\! & 14^2\! & 16^2 \\
      18^3\! & 20^2\! & 22 \\ 24\! & 26\! & 28 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \ \, 3\ \ 5^2 \\ 7^2\ \, 9^2\ 11^3 \\ 13^3\ 15^3 \\
      17^3\, 19^3\, 21^2 \\ 23^2\, 25\ 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 2\! & 4^2\! & 6^3 \\ 8^3\! & 10^3\! & 12^4 \\
      16^4\! & 18^3\! & 20^3 \\ 22^3\! & 24^2\! & 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5\! & 7^2 \\ 9\! & 11^2\! & 13^2 \\ 15^2\! & 17^2\! & 19 \\
      21^2\! & 23\! & 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5^2\! & 7^2 \\ 9^2\! & 11^2\! & 13^3 \\
      15^3\! & 17^2\! & 19^2 \\ 21^2\! & 23^2\! & 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 4\ \ 6 \\ \ 8\ \, 10 \\ 12\ 14^2\, 16 \\
 18\ 20 \\ 22\ 24 \\ \\ \end{smallmatrix} \\
  \hline
 4 &
 \begin{smallmatrix} \,3\ \ 7\ \ 9 \\ 11\ 13 \\ 15\ 17 \\ 19\ 21\ 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 2\ \  4\ \ 6 \\ 8^2\ 10^2 \\ 12^2\, 14^2\, 16^2 \\
       18^2\ 20^2 \\ 22\ 24\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \ \, 3\ \ 5^2 \\ 7^2\ \, 9^2\ 11^3 \\ 13^3\ 15^3 \\
      17^3\, 19^3\, 21^2 \\ 23^2\, 25\ 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 0\! & 2\! & 4^2 \\ 6^3\! & 8^3\! & 10^4 \\
      12^4\! & 14^4\! & 16^4 \\
      18^4\! & 20^3\! & 22^3 \\ 24^2\! & 26\! & 28 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \ 3^2\ \, 5^3 \\ \, 7^4\ \, 9^4\ 11^5 \\ 13^5\ 15^5 \\
      17^5\, 19^4\, 21^4 \\ 23^3\, 25^2\, 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \ 2\ \  4\ \ 6^2 \\ \, 8^2\ 10^2 \\
      12^3\, 14^2\, 16^3 \\
      18^2\ 20^2 \\ 22^2\, 24\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \ 2\ \ \, 4^2\ \, 6^2 \\ \, 8^3\ \, 10^3 \\
      12^3\, 14^4\, 16^3 \\
      18^3\, 20^3\\ 22^2\, 24^2\, 26\, \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5\! & 7 \\ 9^2\! & 11\! & 13^2 \\
      15^2\! & 17\! & 19^2 \\ 21\! & 23\! & 25 \\ \\ \end{smallmatrix} \\
  \hline
 5 &
 \begin{smallmatrix} \, 4\ \ 6\\ \, 8\ \, 10\\
 12\ 14^2\, 16\\ 18\ 20\\ 22\ 24  \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5^2\! & 7^2 \\ 9^2\! & 11^2\! & 13^3 \\
      15^3\! & 17^2\! & 19^2 \\ 21^2\! & 23^2\! & 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 2\! & 4^2\! & 6^3 \\ 8^3\! & 10^3\! & 12^4 \\
      16^4\! & 18^3\! & 20^3 \\ 22^3\! & 24^2\! & 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \ 3^2\ \, 5^3 \\ \, 7^4\ \, 9^4\ 11^5 \\ 13^5\ 15^5 \\
      17^5\, 19^4\, 21^4 \\ 23^3\, 25^2\, 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 0\! & 2^2\! & 4^3 \\ 6^4\! & 8^5\! & 10^6 \\ 12^6\! & 14^6\! & 16^6 \\
      18^6\! & 20^5\! & 22^4 \\ 24^3\! & 26^2\! & 28 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \ 3\ \ 5^2 \\ \, 7^2\ 9^3\ 11^3 \\ 13^3\ 15^3 \\
      17^3\, 19^3\, 21^2 \\ 23^2\, 25\ 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \, 3^2\ \, 5^2 \\ \, 7^3\ 9^4\ 11^4 \\
      13^4\ 15^4 \\
      17^4\, 19^4\, 21^3 \\ 23^2\, 25^2\, 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \ 2\ \ 4\ \ 6 \\ \, 8^2\ 10^2 \\
      12^2\, 14^2\, 16^2 \\
      18^2\ 20^2 \\ 22\ 24\ 26 \\ \\ \end{smallmatrix} \\
  \hline
 6 &
 \begin{smallmatrix} 5 & 9 & 13 \\ 15 & 19 & 23 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 4\ \ 6 \\ \ 8\ \, 10 \\ 12\ 14^2\, 16\\
  18\ 20 \\ 22\ 24 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5\! & 7^2 \\ 9\! & 11^2\! & 13^2 \\ 15^2\! & 17^2\! & 19 \\
      21^2\! & 23\! & 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \ 2\ \ 4\ \ 6^2 \\ 8^2\ 10^2 \\
      12^3\, 14^2\, 16^3 \\
      18^2\ 20^2 \\ 22^2\, 24\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \ 3\ \ 5^2 \\ \, 7^2\ 9^3\ 11^3 \\ 13^3\ 15^3 \\
      17^3\, 19^3\, 21^2 \\ 23^2\, 25\ 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 0\ \ 4\ \ 6 \\ \ 8\ \ 10^2 \\ 12\ 14^2\, 16 \\
      18^2\ 20 \\ 22\ 24\ 28 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 2\ \ 4\ \ 6 \\ 8^2\ 10^2 \\ 12^2\, 14^2\, 16^2 \\
      18^2\ 20^2 \\ 22\ 24\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\ \ 7 \\ \, 9\ \, 11\ \, 13 \\ 15\ 17\ 19 \\ 21\  25 \\ \\ \end{smallmatrix} \\
  \hline
 7 &
 \begin{smallmatrix} \, 5\ \ 7\ \ 11 \\ 13\ 15 \\ 17\ 21\ 23 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \ 4\ \ \ 6^2 \\  8\ \ 10 \\ 12^2\, 14^2\, 16^2 \\
      18\ 20 \\ 22^2\ 24 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5^2\! & 7^2 \\ 9^2\! & 11^2\! & 13^3 \\
      15^3\! & 17^2\! & 19^2 \\ 21^2\! & 23^2\! & 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 2\ \ \, 4^2\ \, 6^2 \\ \, 8^3\ \, 10^3 \\
      12^3\, 14^4\, 16^3 \\
      18^3\ 20^3 \\ 22^2\, 24^2\, 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 1\ \, 3^2\ \, 5^2 \\ \, 7^3\ 9^4\ 11^4 \\
      13^4\ 15^4 \\
      17^4\, 19^4\, 21^3 \\ 23^2\, 25^2\, 27\ \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 2\ \ 4\ \ 6 \\ \, 8^2\ 10^2 \\ 12^2\, 14^2\, 16^2 \\
      18^2\ 20^2 \\ 22\ 24\ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 0\! & 2\! & 4 \\ 6^2\! & 8^2\! & 10^3 \\
      12^3\! & 14^2\! & 16^3 \\ 18^3\! & 20^2\! & 22^2 \\
      24\! & 26\! & 28 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 1\! & 5\! & 7 \\ 9\! & 11^2\! & 13 \\ 15\! & 17^2\! & 19 \\
 21\! & 23\! & 27 \\ \\ \end{smallmatrix} \\
  \hline
 8 &
 \begin{smallmatrix} 6 & 12 \\ 16 & 22 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 5\ \ 7\ \ 11 \\ 13\ 15 \\ 17\ 21\ 23 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 4\ \ \, 6 \\ \ 8\ \, 10 \\ 12\  14^2\, 16 \\ 18\ 20 \\ 22\ 24
      \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\! & 5\! & 7 \\ 9^2\! & 11\! & 13^2 \\
      15^2\! & 17\! & 19^2 \\ 21\! & 23\! & 25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 2 \ \ 4 \ \ 6 \\ 8^2 \ \, 10^2 \\
      12^2\, 14^2\, 16^2 \\
      18^2 \ \, 20^2 \\ 22 \ 24 \ 26 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 3\ \ 7 \\ \, 9\ \, 11\ \, 13 \\ 15\ 17\ 19 \\ 21\  25 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} 1\! & 5\! & 7 \\ 9\! & 11^2\! & 13 \\ 15\! & 17^2\! & 19 \\
 21\! & 23\! & 27 \\ \\ \end{smallmatrix} &
 \begin{smallmatrix} \, 0\ \ 6\ \, 10 \\ 12\ 16 \\ 18\ 22\ 28 \\ \\ \end{smallmatrix} \\
  \hline
\end{array}
\end{equation*}

\vspace*{0.5cm}

\noindent
For each ADE case,
one can easily check the followings.
\begin{enumerate}
 \item
One has $\frak C(k,k')=\frak C(k',k)$ for any $k,k'\in\Pi$.
This implies, for any $k,k'\in\Pi$,
$\HomDz{R}(\ti{M}^k_n,\ti{M}^{k'}_{n'})\simeq
\HomDz{R}(\ti{M}^{k'}_{n'},\ti{M}^k_{n''})$ holds for some
$n,n,n''\in{\mathbb Z}$
such that $\phi^{k}_{n''}-\phi^{k'}_{n'}=\phi^{k'}_{n'}-\phi^k_n$.

 \item A consequence of the Serre duality
(Theorem \ref{thm:grading} (iii)),
\begin{equation*}
\dim_{\mathbb C}(\HomDz{R}(\ti{M}^k_{n},\ti{M}^{k'}_{n'}))
=\dim_{\mathbb C}(\HomDz{R}(\ti{M}^{k'}_{n'},{\mathcal S}(\ti{M}^k_{n})))\ ,
\end{equation*}
can be checked as follows.
For an indecomposable object $\ti{M}^k_n\in\tHMF{R}$,
$k^{\mathcal S}\in\Pi$ denotes the vertex such that
$[F({\mathcal S}(\ti{M}^k_n))]=[M^{k^{\mathcal S}}]\in\Pi$.
Then, for the given multi-set $\frak C(k,k')$, one has
$\frak C(k',k^{\mathcal S})=\{h-2-c\ |\ c\in\frak C(k,k')\}$.

 \item One can check Theorem \ref{thm:grading} (ii-b):
for any two indecomposable objects
$\ti{M}^k_{n},\ti{M}^{k'}_{n'}\in\tHMF{R}$,
$\dim_{\mathbb C}(\HomDz{R}(\ti{M}^k_{n},\ti{M}^{k'}_{n'}))=1$
if $h|\phi^{k'}_{n'}-\phi^k_{n}|=d(k,k')$.
Correspondingly, one has
$\sharp\{c\in\frak C(k,k')\ |\ c=d(\ti{M}^k_{n},\ti{M}^{k'}_{n'})\}=1$.
In addition, the Serre duality implies that
$\dim_{\mathbb C}(\HomDz{R}(\ti{M}^k_{n},\ti{M}^{k'}_{n'}))=1$
if $h|\phi^{k'}_{n'}-\phi^k_{n}|=h-2-d(k^{\mathcal S},k')$.
Namely, one has
$\sharp\{c\in\frak C(k,k')\ |\
c=h-2-d(k^{\mathcal S},k')\}=1$.
These facts, together with Corollary \ref{cor:serre1}, imply that
$\frak C(k,k')=\frak C(k',k)$ is described in the following form:
\begin{equation*}
 \qquad \frak C(k,k')=\{c_1,\cdots, c_s\}\ ,\qquad
 d(k,k')=c_1<c_2\le\cdots\le c_{s-1}< c_s=h-2-d(k^{\mathcal S},k)\
\end{equation*}
for some $s\in{\mathbb Z}_{>0}$.
\end{enumerate}
\end{tbl}

redpajama_set_name: RedPajamaArXiv